%%% -*-BibTeX-*-
%%% ====================================================================
%%% BibTeX-file{
%%% author = "Nelson H. F. Beebe",
%%% version = "1.109",
%%% date = "02 November 2023",
%%% time = "07:26:54 MDT",
%%% filename = "pi.bib",
%%% address = "University of Utah
%%% Department of Mathematics, 110 LCB
%%% 155 S 1400 E RM 233
%%% Salt Lake City, UT 84112-0090
%%% USA",
%%% telephone = "+1 801 581 5254",
%%% FAX = "+1 801 581 4148",
%%% URL = "https://www.math.utah.edu/~beebe",
%%% checksum = "09430 11578 57162 538102",
%%% email = "beebe at math.utah.edu, beebe at acm.org,
%%% beebe at computer.org (Internet)",
%%% codetable = "ISO/ASCII",
%%% keywords = "arctangent; BBP (Bailey, Borwein, Plouffe)
%%% formula; pi calculation; pi computation; PSLQ
%%% algorithm",
%%% license = "public domain",
%%% supported = "yes",
%%% docstring = "This is a bibliography on publications on
%%% the numerical calculation of the fundamental
%%% mathematical constant, pi, the ratio of the
%%% circumference of a circle to its diameter.
%%% It also includes publications about the
%%% mathematical and software algorithms that are
%%% required to tackle large-scale computations
%%% of pi, as well as historical (pre-electronic
%%% computer) work on the problem.
%%%
%%% At version 1.109, the year coverage looked
%%% like this:
%%%
%%% 1727 ( 1) 1826 ( 0) 1925 ( 1)
%%% 1728 ( 0) 1827 ( 0) 1926 ( 2)
%%% 1729 ( 0) 1828 ( 0) 1927 ( 1)
%%% 1732 ( 0) 1831 ( 0) 1930 ( 1)
%%% 1733 ( 0) 1832 ( 0) 1931 ( 1)
%%% 1735 ( 0) 1834 ( 0) 1933 ( 1)
%%% 1737 ( 0) 1836 ( 0) 1935 ( 1)
%%% 1738 ( 0) 1837 ( 0) 1936 ( 1)
%%% 1740 ( 0) 1839 ( 0) 1938 ( 2)
%%% 1741 ( 0) 1840 ( 0) 1939 ( 2)
%%% 1742 ( 0) 1841 ( 0) 1940 ( 1)
%%% 1744 ( 0) 1843 ( 0) 1942 ( 1)
%%% 1747 ( 0) 1846 ( 0) 1945 ( 1)
%%% 1748 ( 0) 1847 ( 0) 1946 ( 2)
%%% 1749 ( 0) 1848 ( 0) 1947 ( 2)
%%% 1750 ( 0) 1849 ( 0) 1948 ( 1)
%%% 1752 ( 0) 1851 ( 0) 1950 ( 6)
%%% 1754 ( 0) 1853 ( 1) 1952 ( 0)
%%% 1756 ( 0) 1855 ( 0) 1954 ( 1)
%%% 1757 ( 0) 1856 ( 0) 1955 ( 4)
%%% 1759 ( 0) 1858 ( 0) 1957 ( 1)
%%% 1760 ( 0) 1859 ( 0) 1958 ( 1)
%%% 1761 ( 0) 1860 ( 0) 1959 ( 1)
%%% 1762 ( 0) 1861 ( 0) 1960 ( 2)
%%% 1763 ( 0) 1862 ( 0) 1961 ( 1)
%%% 1764 ( 0) 1863 ( 0) 1962 ( 4)
%%% 1767 ( 0) 1866 ( 0) 1965 ( 1)
%%% 1768 ( 1) 1867 ( 0) 1966 ( 1)
%%% 1769 ( 0) 1868 ( 0) 1967 ( 3)
%%% 1770 ( 0) 1869 ( 0) 1968 ( 1)
%%% 1771 ( 0) 1870 ( 0) 1969 ( 3)
%%% 1772 ( 0) 1871 ( 2) 1970 ( 2)
%%% 1773 ( 0) 1872 ( 0) 1971 ( 2)
%%% 1774 ( 0) 1873 ( 1) 1972 ( 1)
%%% 1775 ( 0) 1874 ( 0) 1973 ( 1)
%%% 1778 ( 0) 1877 ( 0) 1976 ( 5)
%%% 1779 ( 0) 1878 ( 0) 1977 ( 1)
%%% 1780 ( 0) 1879 ( 1) 1978 ( 3)
%%% 1781 ( 0) 1880 ( 0) 1979 ( 3)
%%% 1782 ( 0) 1881 ( 0) 1980 ( 2)
%%% 1783 ( 0) 1882 ( 1) 1981 ( 3)
%%% 1784 ( 0) 1883 ( 1) 1982 ( 2)
%%% 1785 ( 0) 1884 ( 0) 1983 ( 3)
%%% 1786 ( 0) 1885 ( 0) 1984 ( 4)
%%% 1787 ( 0) 1886 ( 0) 1985 ( 3)
%%% 1788 ( 0) 1887 ( 0) 1986 ( 7)
%%% 1789 ( 0) 1888 ( 0) 1987 ( 4)
%%% 1790 ( 0) 1889 ( 0) 1988 ( 8)
%%% 1791 ( 0) 1890 ( 0) 1989 ( 6)
%%% 1792 ( 0) 1891 ( 1) 1990 ( 4)
%%% 1793 ( 0) 1892 ( 0) 1991 ( 4)
%%% 1794 ( 0) 1893 ( 0) 1992 ( 4)
%%% 1795 ( 0) 1894 ( 0) 1993 ( 4)
%%% 1796 ( 0) 1895 ( 1) 1994 ( 6)
%%% 1797 ( 0) 1896 ( 1) 1995 ( 4)
%%% 1798 ( 0) 1897 ( 0) 1996 ( 6)
%%% 1799 ( 0) 1898 ( 0) 1997 ( 14)
%%% 1800 ( 0) 1899 ( 0) 1998 ( 7)
%%% 1801 ( 0) 1900 ( 0) 1999 ( 6)
%%% 1802 ( 0) 1901 ( 0) 2000 ( 10)
%%% 1803 ( 0) 1902 ( 0) 2001 ( 6)
%%% 1804 ( 0) 1903 ( 0) 2002 ( 4)
%%% 1805 ( 0) 1904 ( 1) 2003 ( 5)
%%% 1806 ( 0) 1905 ( 0) 2004 ( 9)
%%% 1807 ( 0) 1906 ( 0) 2005 ( 5)
%%% 1808 ( 0) 1907 ( 0) 2006 ( 6)
%%% 1809 ( 0) 1908 ( 0) 2007 ( 1)
%%% 1810 ( 0) 1909 ( 0) 2008 ( 9)
%%% 1811 ( 0) 1910 ( 0) 2009 ( 3)
%%% 1812 ( 0) 1911 ( 0) 2010 ( 13)
%%% 1813 ( 0) 1912 ( 0) 2011 ( 16)
%%% 1814 ( 0) 1913 ( 0) 2012 ( 6)
%%% 1815 ( 0) 1914 ( 1) 2013 ( 17)
%%% 1816 ( 0) 1915 ( 0) 2014 ( 6)
%%% 1817 ( 0) 1916 ( 0) 2015 ( 7)
%%% 1818 ( 0) 1917 ( 0) 2016 ( 2)
%%% 1819 ( 0) 1918 ( 0) 2017 ( 4)
%%% 1820 ( 0) 1919 ( 0) 2018 ( 3)
%%% 1821 ( 0) 1920 ( 0) 2019 ( 2)
%%% 1822 ( 0) 1921 ( 1) 2020 ( 6)
%%% 1823 ( 0) 1922 ( 0) 2021 ( 1)
%%% 1824 ( 0) 1923 ( 0) 2022 ( 3)
%%% 1825 ( 0) 1924 ( 1) 2023 ( 3)
%%%
%%% Article: 222
%%% Book: 36
%%% InBook: 1
%%% InCollection: 8
%%% InProceedings: 11
%%% Misc: 19
%%% Proceedings: 5
%%% TechReport: 12
%%% Unpublished: 19
%%%
%%% Total entries: 333
%%%
%%% Despite its representation by a Greek letter,
%%% the Greeks did not use that symbol for the
%%% constant. Instead, it was Leonhard Euler in
%%% September 1727 who first used the name pi for
%%% the ratio of the periphery of a circle to its
%%% radius ($ 2 \pi $ in modern notation); see
%%% entry Euler:1727:TEP. He later used it for
%%% the ratio of the periphery to the diameter,
%%% and that convention was soon widely adopted.
%%%
%%% The constant pi was proved to be irrational
%%% by Lambert in 1766, using a continued
%%% fraction, and thus showing that the digits of
%%% pi neither terminate, nor repeat in any
%%% number base (other than pi itself, or
%%% rational multiples thereof).
%%%
%%% In 1882, Lindemann proved that pi is also
%%% transcendental, showing that the digits of an
%%% integer polynomial of pi cannot repeat, and
%%% thus, nonzero positive integral powers of pi
%%% cannot have repeating decimals.
%%%
%%% Human interest in the problem of calculating
%%% numerical values of pi has existed for more
%%% than 1500 years, but it was only the advent
%%% of electronic digital computers that made it
%%% possible to advance beyond a few hundred
%%% known digits. By mid-2010, the record for
%%% correct decimal digits of pi stood at about 5
%%% * 10**12, and by late 2011, that had grown to
%%% more than 10**13 (10 trillion) decimal
%%% digits. See entries Bailey:2011:CPI and
%%% Bailey:2013:PDU for tables of historical,
%%% early computer, and modern computer records
%%% for the digits of pi, and entries
%%% Yee:2013:IST, Yee:2017:PNL, Yee:2020:CMT, and
%%% Yee:2022:CMT for the latest records. See
%%% entry Shelburne:2012:ED for a reconstruction
%%% of the first computer calculation of pi and e
%%% (about 2000 decimal digits each), carried out
%%% on the ENIAC on Labor Day (early September)
%%% weekend, 1949.
%%%
%%% In 1997, a remarkable equation, the
%%% now-famous BBP (Bailey, Borwein, and Plouffe)
%%% formula was discovered. In (La)TeX markup
%%% that produces a one-line typeset equation, it
%%% can be stated like this:
%%%
%%% \pi = \sum_{k = 0}^\infty
%%% \frac{1}{16^k}
%%% \left (
%%% \frac{4}{8 k + 1} -
%%% \frac{2}{8 k + 4} -
%%% \frac{1}{8 k + 5} -
%%% \frac{1}{4 k + 6}
%%% \right )
%%%
%%% The BBP discoverers showed that their formula
%%% has the astonishing property that it can be
%%% used to generate digits of pi in any base
%%% that is a power of 2, STARTING from the n-th
%%% digit, and WITHOUT knowing all previous
%%% digits 1, 2, ..., n - 1.
%%%
%%% It has since been proved that no such formula
%%% exists for pi in base 10, and that similar
%%% formulas can be exhibited for other
%%% constants, such as \pi^2, \zeta(2), \zeta(3),
%%% Catalan's constant, \log(k) (k in [2, 22]),
%%% and many arctangents.
%%%
%%% By contrast, it is conjectured that no such
%%% formulas exist for the base of the natural
%%% logarithm, e = \exp(1) ~= 2.718281828....
%%%
%%% A long-standing, but still unproved,
%%% conjecture, understandable even to a grade
%%% school student, is that the digits of pi form
%%% a random sequence: that is, in a sufficiently
%%% large digit sequence, the digits each occur
%%% with equal probability. Such a number is
%%% called a ``normal number''. Note that this
%%% does NOT mean that short digit sequences are
%%% random: the sequences 0123456789 and
%%% 7777777777 both occur within the first
%%% 22,900,000,000 decimal digits of pi. The
%%% six-digit sequence 999999 appears at the 762nd
%%% decimal place, and is called the ``Feynman
%%% point'', after Physics Nobel laureate Richard
%%% Feynman: for background, see
%%%
%%% http://en.wikipedia.org/wiki/Feynman_point
%%%
%%% Normality has been proven for some other
%%% irrational constants, but never for pi.
%%% Statistical analysis of the known computed
%%% digits of pi strongly suggest normality, but
%%% a mathematical proof remains elusive, and
%%% appears at present to be very difficult.
%%%
%%% See entries Marsaglia:2005:RPO and
%%% Marsaglia:2006:RCS for remarks on statistical
%%% measures of the randomness of digits of pi,
%%% and how many such proposed measures are
%%% seriously flawed. The second of those
%%% articles concludes with this remark about
%%% tests of randomness: ``$\pi$ sails through
%%% all of them''. See also the more recent
%%% paper Ganz:2014:DES and its reproduction, and
%%% refutation of statistics, in Bailey:2016:RCS.
%%%
%%% Web sites about pi include:
%%%
%%% http://carma.newcastle.edu.au/bbp
%%% http://carma.newcastle.edu.au/jon/bio_short.html
%%% http://carma.newcastle.edu.au/jon/pi-2010.pdf
%%% http://crd.lbl.gov/~dhbailey/pi/
%%% http://www.experimentalmath.info/
%%%
%%% The checksum field above contains a CRC-16
%%% checksum as the first value, followed by the
%%% equivalent of the standard UNIX wc (word
%%% count) utility output of lines, words, and
%%% characters. This is produced by Robert
%%% Solovay's checksum utility.",
%%% }
%%% ====================================================================
@Preamble{
"\def \cprime {$'$}" #
"\ifx \undefined \arccot \def \arccot{{\rm arccot}} \fi" #
"\ifx \undefined \booktitle \def \booktitle #1{{{\em #1}}} \fi" #
"\ifx \undefined \mathbb \def \mathbb #1{{\bf #1}}\fi" #
"\ifx \undefined \mathbf \def \mathbf #1{{\bf #1}}\fi" #
"\ifx \undefined \mathrm \def \mathrm #1{{\rm #1}}\fi"
}
%%% ====================================================================
%%% Acknowledgement abbreviations:
@String{ack-nhfb = "Nelson H. F. Beebe,
University of Utah,
Department of Mathematics, 110 LCB,
155 S 1400 E RM 233,
Salt Lake City, UT 84112-0090, USA,
Tel: +1 801 581 5254,
FAX: +1 801 581 4148,
e-mail: \path|beebe@math.utah.edu|,
\path|beebe@acm.org|,
\path|beebe@computer.org| (Internet),
URL: \path|https://www.math.utah.edu/~beebe/|"}
%%% ====================================================================
%%% Institute abbreviations:
%%% ====================================================================
%%% Journal abbreviations:
@String{j-ACM-COMM-COMP-ALGEBRA = "ACM Communications in Computer Algebra"}
@String{j-ADV-DIFFERENCE-EQU = "Advances in Difference Equations"}
@String{j-AMER-MATH-MONTHLY = "American Mathematical Monthly"}
@String{j-AMER-STAT = "The American Statistician"}
@String{j-APPL-MATH-COMP = "Applied Mathematics and Computation"}
@String{j-ARCH-HIST-EXACT-SCI = "Archive for History of Exact Sciences"}
@String{j-BIT = "BIT"}
@String{j-BRITISH-J-HIST-SCI = "British Journal for the History of Science"}
@String{j-BULL-AMS = "Bulletin of the American Mathematical Society"}
@String{j-BULL-AMS-N-S = "Bulletin of the American Mathematical Society
(new series)"}
@String{j-CACM = "Communications of the ACM"}
@String{j-CAN-J-MATH = "Canadian Journal of Mathematics = Journal
canadien de math{\'e}matiques"}
@String{j-CAN-MATH-BULL = "Bulletin canadien de math{\'e}matiques =
Canadian Mathematical Bulletin"}
@String{j-CHIFFRES = "Chiffres: Revue de l'Association
fran{\c{c}}aise de Calcul"}
@String{j-COLLOQ-MATH = "Colloquium Mathematicum"}
@String{j-COMP-J = "The Computer Journal"}
@String{j-COMP-PHYS-COMM = "Computer Physics Communications"}
@String{j-COMPUT-MATH-APPL = "Computers and Mathematics with Applications"}
@String{j-COMPUT-SCI-ENG = "Computing in Science and Engineering"}
@String{j-COMPUTING = "Computing"}
@String{j-EULERIANA = "Euleriana"}
@String{j-EXP-MATH = "Experimental mathematics"}
@String{j-FIB-QUART = "Fibonacci Quarterly"}
@String{j-HIST-MATH = "Historia Mathematica"}
@String{j-IEEE-ANN-HIST-COMPUT = "IEEE Annals of the History of Computing"}
@String{j-IEEE-SPECTRUM = "IEEE Spectrum"}
@String{j-INT-J-MATH-EDU-SCI-TECH = "International Journal of Mathematical
Education in Science and Technology"}
@String{j-INT-J-MOD-PHYS-C = "International Journal of Modern Physics C
[Physics and Computers]"}
@String{j-INT-J-NUMBER-THEORY = "International Journal of Number Theory (IJNT)"}
@String{j-INT-J-PARALLEL-PROG = "International Journal of Parallel
Programming"}
@String{j-J-ALG = "Journal of Algorithms"}
@String{j-J-ACM = "Journal of the ACM"}
@String{j-J-AUTOM-REASON = "Journal of Automated Reasoning"}
@String{j-J-COMPUT-APPL-MATH = "Journal of Computational and Applied
Mathematics"}
@String{j-J-DIFFERENCE-EQU-APPL = "Journal of Difference Equations and
Applications"}
@String{j-J-MATH-PHYS = "Journal of Mathematical Physics"}
@String{j-J-NUMER-METHODS-COMPUT-APPL = "Journal on Numerical Methods and
Computer Applications"}
@String{j-J-R-STAT-SOC-SER-A-GENERAL = "Journal of the Royal Statistical
Society. Series A (General)"}
@String{j-J-REINE-ANGEW-MATH = "Journal f{\"u}r die reine und angewandte
Mathematik"}
@String{j-J-STAT-COMPUT-SIMUL = "Journal of Statistical Computation and
Simulation"}
@String{j-J-SUPERCOMPUTING = "The Journal of Supercomputing"}
@String{j-J-UCS = "J.UCS: Journal of Universal Computer
Science"}
@String{j-MATH-ANN = "Mathematische Annalen"}
@String{j-MATH-COMPUT = "Mathematics of Computation"}
@String{j-MATH-GAZ = "Mathematical Gazette"}
@String{j-MATH-INTEL = "The Mathematical Intelligencer"}
@String{j-MATH-MAG = "Mathematics Magazine"}
@String{j-MATH-TABLES-OTHER-AIDS-COMPUT = "Mathematical Tables and Other Aids
to Computation"}
@String{j-MATH-TEACH = "The Mathematics Teacher"}
@String{j-NAMS = "Notices of the American Mathematical
Society"}
@String{j-NUMER-ALGORITHMS = "Numerical Algorithms"}
@String{j-OSIRIS = "Osiris"}
@String{j-PAC-J-MATH = "Pacific Journal of Mathematics"}
@String{j-PARALLEL-COMPUTING = "Parallel Computing"}
@String{j-PHYS-REV-A = "Physical Review A (Atomic, Molecular,
and Optical Physics)"}
@String{j-PROC-AM-MATH-SOC = "Proceedings of the American Mathematical
Society"}
@String{j-PROC-NATL-ACAD-SCI-USA = "Proceedings of the {National Academy of
Sciences of the United States of America}"}
@String{j-PROC-R-SOC-LOND = "Proceedings of the Royal Society of London"}
@String{j-SANKHYA-B = "Sankhy{\={a}} (Indian Journal of Statistics),
Series B. Methodological"}
@String{j-SCI-AMER = "Scientific American"}
@String{j-SCI-COMPUT = "Scientific Computing"}
@String{j-SCI-EDUC-SPRINGER = "Science \& Education (Springer)"}
@String{j-SCIENCE-NEWS = "Science News (Washington, DC)"}
@String{j-SIAM-J-COMPUT = "SIAM Journal on Computing"}
@String{j-SIGNUM = "ACM SIGNUM Newsletter"}
@String{j-STATISTICS = "Statistics: a Journal of Theoretical
and Applied Statistics"}
@String{j-TOMS = "ACM Transactions on Mathematical Software"}
@String{j-TRANS-INFO-PROCESSING-SOC-JAPAN = "Transactions of the Information
Processing Society of Japan"}
%%% ====================================================================
%%% Publishers and their addresses:
@String{pub-A-K-PETERS = "A. K. Peters, Ltd."}
@String{pub-A-K-PETERS:adr = "Wellesley, MA, USA"}
@String{pub-ACADEMIC = "Academic Press"}
@String{pub-ACADEMIC:adr = "New York, NY, USA"}
@String{pub-AMS = "American Mathematical Society"}
@String{pub-AMS:adr = "Providence, RI, USA"}
@String{pub-BARNES-NOBLE = "Barnes and Noble"}
@String{pub-BARNES-NOBLE:adr = "New York, NY, USA"}
@String{pub-BASIC-BOOKS = "Basic Books"}
@String{pub-BASIC-BOOKS:adr = "New York, NY, USA"}
@String{pub-CAMBRIDGE = "Cambridge University Press"}
@String{pub-CAMBRIDGE:adr = "Cambridge, UK"}
@String{pub-CLARENDON = "Clarendon Press"}
@String{pub-CLARENDON:adr = "Oxford, UK"}
@String{pub-GOLEM = "Golem Press"}
@String{pub-GOLEM:adr = "Boulder, CO, USA"}
@String{pub-IEEE = "IEEE Computer Society Press"}
@String{pub-IEEE:adr = "1109 Spring Street, Suite 300, Silver Spring,
MD 20910, USA"}
@String{pub-LITTLE-BROWN = "Little, Brown and Company"}
@String{pub-LITTLE-BROWN:adr = "Boston, Toronto, London"}
@String{pub-MATH-ASSOC-AMER = "Mathematical Association of America"}
@String{pub-MATH-ASSOC-AMER:adr = "Washington, DC, USA"}
@String{pub-PLENUM = "Plenum Press"}
@String{pub-PLENUM:adr = "New York, NY, USA; London, UK"}
@String{pub-PRINCETON = "Princeton University Press"}
@String{pub-PRINCETON:adr = "Princeton, NJ, USA"}
@String{pub-PROMETHEUS-BOOKS = "Prometheus Books"}
@String{pub-PROMETHEUS-BOOKS:adr = "Amherst, NY, USA"}
@String{pub-SIAM = "Society for Industrial and Applied
Mathematics"}
@String{pub-SIAM:adr = "Philadelphia, PA, USA"}
@String{pub-ST-MARTINS = "St. Martin's Press"}
@String{pub-ST-MARTINS:adr = "New York, NY, USA"}
@String{pub-SV = "Spring{\-}er-Ver{\-}lag"}
@String{pub-SV:adr = "Berlin, Germany~/ Heidelberg,
Germany~/ London, UK~/ etc."}
@String{pub-W-H-FREEMAN = "W. H. {Freeman and Company}"}
@String{pub-W-H-FREEMAN:adr = "New York, NY, USA"}
@String{pub-WI = "Wiley-In{\-}ter{\-}sci{\-}ence"}
@String{pub-WI:adr = "New York, NY, USA"}
%%% ====================================================================
%%% Series abbreviations:
@String{ser-LNCS = "Lecture Notes in Computer Science"}
%%% ====================================================================
%%% Bibliography entries, sorted by ascending year, and then by citation
%%% label, with ``bibsort --byyear'':
@Article{Euler:1727:TEP,
author = "Leonhard Euler",
title = "Testamen explicationis phaenomenorum aeris. ({Latin})
[{An} Essay Explaining the Properties of Air]",
journal = "Comm. Ac. Scient. Petr.",
volume = "2",
pages = "347--368",
month = sep,
year = "1727",
bibdate = "Mon Jun 10 08:47:38 2013",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
note = "Translation to English, and annotations, by Ian
Bruce.",
URL = "http://17centurymaths.com/contents/euler/e007tr.pdf",
acknowledgement = ack-nhfb,
language = "Latin",
remark = "This is the paper in which Euler used the Greek letter
pi for the ratio of the periphery of a circle to its
radius ($ 2 \pi $ in modern notation). Euler later used
the same symbol for the ratio of the periphery to the
diameter, and that convention was soon widely
adopted.",
}
@Article{Lambert:1768:MQP,
author = "Johann Heinrich Lambert",
title = "{M{\'e}moire} sur quelques propri{\'e}t{\'e}s
remarquables des quantit{\'e}s transcendentes
circulaires et logarithmiques. ({French}) [{Note} on
some remarkable properties of circular and logarithmic
transcendental quantities]",
journal = "Histoire de {l'Acad{\'e}mie (Berlin)}",
volume = "XVII",
pages = "265--322",
month = "????",
year = "1768",
bibdate = "Sat Apr 23 10:07:00 2011",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
note = "In this famous paper, Lambert proved that $ \pi $ is
irrational. See \cite{Laczkovich:1997:LPI} for further
remarks, a simplification of the proof, and references
to earlier papers that discuss Lambert's proof.",
acknowledgement = ack-nhfb,
fjournal = "Histoire de {l'Acad{\'e}mie (Berlin)}",
language = "French",
remark = "One Web source says the paper is from 1761, but only
printed in 1768. The continued fraction in a
low-resolution image of an equation on page 288 of the
paper appears to be $ \tan (\phi / \omega) = \phi /
(\omega - \phi \phi / (3 \omega - \phi \phi / (5 \omega
- \phi \phi / (7 \omega - \phi \phi / (9 \omega -
\mathrm {etc.}))))) $. In modern terms, this can be
written as $ \tan (x) = x / (1 - x^2 / (3 - x^2 / (5 -
x^2 / (7 - x^2 / (9 - \mathrm {etc.}))))) $. Lambert
proved that continued fraction expansion, then showed
that if $x$ is nonzero and rational, then the continued
fraction must be irrational. Because $ \tan (\pi / 4) =
1$, it follows that $ \pi / 4$ is irrational, and
therefore, $ \pi $ is irrational.",
}
@Book{Shanks:1853:CMC,
author = "W. Shanks",
title = "Contributions to Mathematics, Comprising Chiefly of
the Rectification of the Circle to 607 Places of
Decimals",
publisher = "G. Bell",
address = "London, UK",
pages = "xvi + 95 + 1",
year = "1853",
LCCN = "QA467 .S53 1853",
bibdate = "Tue Apr 26 15:55:02 2011",
bibsource = "fsz3950.oclc.org:210/WorldCat;
https://www.math.utah.edu/pub/tex/bib/pi.bib;
library.ox.ac.uk:210/ADVANCE",
acknowledgement = ack-nhfb,
remark = "Reprinted in: Mathematics, 1850--1910, in the
Mathematics Collection, Brown University Library. Reel
no. 7420. Item no. 1. Reproduced for the Great
Collections Microfilming Project, Phase II, Research
Libraries Group.",
subject = "circle-squaring; pi; mathematics; geometry",
}
@Article{Frisby:1871:C,
author = "E. Frisby",
title = "On the calculation of $ \pi $",
journal = "Messenger (2)",
volume = "II",
number = "??",
pages = "114--114",
month = "????",
year = "1871",
bibdate = "Mon Apr 25 18:00:24 2011",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
ZMnumber = "04.0255.02",
acknowledgement = ack-nhfb,
classmath = "*51M04 (Elementary problems in Euclidean geometries)",
keywords = "$\pi$",
reviewer = "Glaisher, Prof. (Cambridge) (Ohrtmann, Dr. (Berlin))",
}
@Article{Glaisher:1871:RC,
author = "J. W. L. Glaisher",
title = "Remarks on the calculation of $ \pi $",
journal = "Messenger (2)",
volume = "II",
number = "??",
pages = "119--128",
month = "????",
year = "1871",
bibdate = "Mon Apr 25 17:40:04 2011",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
ZMnumber = "04.0255.04",
abstract = "{Die Bemerkungen am Anfange der Arbeit beziehen sich
auf die beiden obigen Arbeiten (JFM 04.0255.02 und JFM
04.0255.03). Herr Glaisher berichtet {\"u}ber Versuche,
{\"a}hnlich denen des Herrn Fox, die 1855 auf
Veranlassung de Morgan's von Herrn Ambroise Smith
gemacht worden sind. Er bemerkt, dass die von Herrn
Frisby benutzten Reihen unabh{\"a}ngig von einander
gegeben worden sind von Hutton, Euler, H. James
Thomson, Blissard und de Morgan, und discutirt einige
{\"a}hnliche Reihen von Euler und Hutton. Dann folgt
eine Liste der Berechner von $ \pi $ und der von ihnen
erreichten Stellenzahl, von Archimedes bis zur
Jetztzeit. Diese Liste beruht auf einer {\"a}hnlichen,
die Herr Bierens de Haan in den ``Verhandlingen'' von
Amsterdam, Bd. IV. p. 22 1858 gegeben hat. Dieselbe
zeigt das allm{\"a}lige Wachsen der mathematischen
H{\"u}lfsmittel im Verlaufe von 2000 Jahren. Der
{\"u}brige Theil der Arbeit ist haupts{\"a}chlich den
Werken und Rechnungen von Ludolf van Ceulen und Snell
gewidmet. Der Verfasser bringt Gr{\"u}nde f{\"u}r die
Vermuthung vor, dass van Ceulen's Werth mit 35 Stellen
zuerst durch die Worte auf seinem Grabe bekannt wurden.
(Zus{\"a}tze und Verbesserungen zu der Arbeit und zu
der Liste finden sich in des Verfassers Arbeit: ``On
the quadrature of the circle, A. D. 1580-1630.''
Messenger (2) III., siehe den folgenden Band dieses
Jahrbuches.)}",
acknowledgement = ack-nhfb,
classmath = "{*51M04 (Elementary problems in Euclidean
geometries)}",
keywords = "{$\pi$}",
language = "English",
reviewer = "{Glaisher, Prof. (Cambridge) (Ohrtmann, Dr.
(Berlin))}",
}
@Article{Shanks:1873:ENV,
author = "William Shanks",
title = "On the Extension of the Numerical Value of $ \pi $",
journal = j-PROC-R-SOC-LOND,
volume = "21",
number = "??",
pages = "315--319",
day = "15",
month = may,
year = "1873",
CODEN = "PRSLAZ",
ISSN = "0370-1662",
bibdate = "Fri Jul 01 06:48:41 2011",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
URL = "http://www.jstor.org/stable/113051",
acknowledgement = ack-nhfb,
fjournal = "Proceedings of the Royal Society of London",
remark = "From the first page: ``The values of $ \tan^{-1}(1 /
5) $ and of $ \tan^{-1} $ are each given below to 709,
and the value of $ \pi $ to 707 decimals. It will be
observed that a few figures in the values of $
\tan^{-1}(1 / 5) $ and of $ \pi $, published in 1853,
were erroneous. The author detected the error quite
recently, and has corrected it. \ldots{} Prof. Richter,
of Elbing, found $ \pi $ to 500 decimals in the year
1853---all of which agree with the author's, published
early in the same year.''",
}
@Article{Polster:1879:NIS,
author = "F. Polster",
title = "A new infinite series, which is very convenient for
the computation of $ \pi $",
journal = "J. Blair Bl.",
volume = "XV",
number = "??",
pages = "155--158",
month = "????",
year = "1879",
bibdate = "Mon Apr 25 17:54:07 2011",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
ZMnumber = "11.0181.01",
acknowledgement = ack-nhfb,
classmath = "*40A05 (Convergence of series and sequences) 40A25
(Approximation to limiting values) 41A58 (Series
expansions)",
keywords = "approximation of $\pi$; series expansion",
language = "German",
reviewer = "G{\"u}nther, Prof. (Ansbach)",
xxtitle = "{Eine neue unendliche Reihe, welche zur Berechnung der
Ludolphine sehr bequem ist}",
}
@Article{vonLindemann:1882:ZGN,
author = "Carl Louis Ferdinand von Lindemann",
title = "{{\"U}ber die Zahl $ \pi $}. ({German}) [{On} the
number $ \pi $]",
journal = j-MATH-ANN,
volume = "20",
number = "??",
pages = "213--225",
month = "????",
year = "1882",
CODEN = "MAANA3",
ISSN = "0025-5831 (print), 1432-1807 (electronic)",
ISSN-L = "0025-5831",
bibdate = "Sat Apr 23 10:13:07 2011",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
note = "In this famous paper, von Lindemann proved that $ \pi
$ is transcendental, showing that it is impossible to
square the circle by compass and straightedge, a
problem dating back before 400 BCE in Greece.",
ZMnumber = "FM 14.0369.04",
abstract = "In seiner Abhandlung: Sur la fonction exponentielle
(C. R. Bd. LXXVII, s. F. d. M. V. (1873) p. 248, JFM
05.0248.01) hat Herr Hermite die Unm{\"o}glichkeit
einer Relation von der Form $$ N_0 e^{z_0} + N_1
e^{z_1} + \cdots + N_n e^{z_n} = 0 $$ bewiesen, wo
sowohl die $z$ als die $N$ als ganz vorausgesetzt
werden. Herr Lindemann (siehe auch JFM 14.0369.02, JFM
14.0369.03) erweitert die hier gemachten Schl{\"u}sse
und gelangt zu folgendem Satze: ``Sind $$ f_1 (z) = 0,
f_2 (z) = 0, \ldots, f_s(z) = 0 $$ $s$ algebraische
Gleichungen, von denen jede irreductibel und von der
Form $$ z^n + a_1 z^{n - 1} \ldots + a_n = 0 $$ ist, wo
unter $ a_1$, $ a_2$, $ \ldots $, $ a_n$ ganze Zahlen
zu verstehen sind, werden ferner mit $ z_i$, $ z_i'$, $
z_i''$, $ \ldots $ die Wurzeln der Gleichung $ f_i(z) =
0$ bezeichnet, wird kurz $$ \varsigma e^{z_i} = e^{z_i}
+ e^{z_i '} + e^{z_i ''} + \ldots $$ gesetzt, bedeuten
endlich $ N_0$, $ N_1$, $ \ldots $, $ N_s$ beliebige
ganze Zahlen, welche nicht s{\"a}mmtlich gleich Null
sind, so kann eine Relation von der Form $$ 0 = N_0 +
N_1 \varsigma e^{z_1} + N_2 \varsigma e^{z_2} + \cdots
+ N_s \varsigma e^{z_s} $$ nicht bestehen, es sei denn,
dass eine der Gr{\"o}ssen $z$ gleich Null
ist.''\par
Ersetzt man die Gleichungen $ f_i(z) = 0$ durch
diejenigen irreduciblen Gleichungen, welche bez. von
den Zahlen $$ Z_1 = z_1, Z_2 = z_1 + z_2, Z_3 = z_1 +
z_2 + z_3, \ldots, Z_n = z_1 + z_2 \cdots + z_n $$
befriedigt werden, so f{\"u}hrt dieser besondere Fall
zu dem Satze: ``Ist $z$ eine von Null verschiedene
rationale oder algebraisch irrationale Zahl, so ist $
e^{\tau }$ immer transcendent.'' Damit ist bewiesen,
dass die Ludolph'sche Zahl $ \pi $ eine transcendente
Zahl ist. Die angef{\"u}hrten S{\"a}tze bleiben
bestehen, wenn man unter den $ N_i$ nicht ganze oder
rationale, sondern beliebige algebraisch-irrationale
Zahlen versteht. Analog folgt aus dem obigen Satze der
folgende: ``Versteht man unter $ N_0$, $ N_1$, $ \ldots
$, $ N_n$ beliebige, und unter $ z_0$, $ z_1$, $ \ldots
$, $ z_n$ beliebige, von einander verschiedene (reelle
oder complexe) algebraische Zahlen, so kann eine
Relation von der Form $$ 0 = N_0 e^{z_0} + N_1 e^{z_1}
+ \cdots + N_n e^{z_n} $$ nicht bestehen, es sei denn,
dass die $ N_i$ s{\"a}mmtlich gleich Null werden.''",
abstract-2 = "In his paper {\em Sur la fonction exponential} (C.R.
Bd. LXXVII, S.F.D. M.V. (1873) p. 248, JFM 05.0248.01)
Mr. Hermite has proved the impossibility of a relation
of the form $$N_0 e^{z_0} + N_1 e^{z_1} + \cdots + N_n
e^{z_n} = 0$$, where both $z$ and $N$ are given. Mr.
Lindemann (see also JFM 14.0369.02, JFM 14.0369.03)
extends the conclusions made here and arrives at the
following sentence: ``If $$f_1 (z) = 0, f_2 (z) = 0,
\ldots, f_s (z) = 0$$ $s$ are irreducible algebraic
equations of the form $$z^{n} + a_1z^{n-1} \ldots + a_n
= 0$$, where $a_1$, $a_2$, $\ldots$ $a_n$ are whole
numbers, and $z_i$, $z_i'$, $z_i''$, $\ldots$ are roots
of the equation $f_i(z) = 0$, and $$\varsigma e^{z_i} =
e^{z_i} + e^{z_i '} + e^{z_i ' '} + \ldots$$, where
$N_0$, $N_1$, $\ldots$, $N_s$ are arbitrary nonzero
whole numbers, then a relation of the form $$0 = N_0 +
N_1 \varsigma e^{z_1} + N_2 \ varSigma e^{z_2} + \cdots
+ N_s \varsigma e^{z_s}$$ does not exist, unless one of
the values $z$ is zero.\par
If one replaces the equations $f_i (z) = 0$ by those
irreducible equations for which the numbers $$Z_1 =
z_1, Z_2 = z_1 + z_2, Z_3 = z_1 + z_2 + z_3, \ldots,
Z_n = z_1 + z_2 \cdots + z_n$$ are satisfied, then this
is a special case of the sentence: ``If $z$ one of zero
different rational or algebraically irrational numbers,
then $e^{\tau}$ is always transcendental. ``Thus, it is
proven that the Ludolph number of $\pi$ is a
transcendental number. The aforementioned theorem
holds, if one of the $N_i$ is not whole or rational,
but instead, is an arbitrary algebraic-irrational
number. It similarly follows from the above statement
that: `One concludes that if $N_0$, $N_1$, $\ldots$,
$N_n$ are arbitrary, and if $z_0$, $z_1$, $\ldots$,
$z_n$ are arbitrary, different (real or complexe)
algebraic numbers, then a relation of the form $$0 =
N_0e^{z_0} + N_1e^{z_1} + \cdots + N_n e^{z_n}$$ cannot
exist, unless $N_i$ is zero",
acknowledgement = ack-nhfb,
fjournal = "Mathematische Annalen",
language = "German",
remark = "Improve the crude English translation of the
abstract!",
xxjournal = "Klein Ann.",
}
@Article{Glaisher:1883:CHL,
author = "J. W. L. Glaisher",
title = "Calculation of the hyperbolic logarithm of $ \pi $",
journal = "J. Lond. M. S. Proc.",
volume = "XIV",
number = "??",
pages = "134--139",
month = "????",
year = "1883",
bibdate = "Mon Apr 25 17:40:04 2011",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
ZMnumber = "15.0997.04",
abstract = "Berechnung auf zwei Weisen und Vergleich verschiedener
Methoden und Resultate. [Computation two ways and
comparison of different methods and results.]",
acknowledgement = ack-nhfb,
language = "English",
reviewer = "{Ohrtmann, Dr. (Berlin)}",
}
@Article{Glaisher:1891:CHL,
author = "J. W. L. Glaisher",
title = "Calculation of the hyperbolic logarithm of $ \pi $ to
thirty decimal places --- Addition to the paper",
journal = "Quart. J.",
volume = "XXV",
number = "??",
pages = "362--368, 384",
month = "????",
year = "1891",
MRclass = "33F05",
bibdate = "Mon Apr 25 17:40:04 2011",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
ZMnumber = "23.0277.01",
acknowledgement = ack-nhfb,
classmath = "*33F05 (Numerical approximation of special
functions)",
keywords = "Calculation of $\log\pi$",
language = "English",
reviewer = "Weltzien, Dr. (Zehlendorf)",
}
@Article{Smith:1895:HSA,
author = "David Eugene Smith",
title = "Historical Survey of the Attempts at the Computation
and Construction of $ \pi $",
journal = j-AMER-MATH-MONTHLY,
volume = "2",
number = "12",
pages = "348--351",
month = dec,
year = "1895",
CODEN = "AMMYAE",
ISSN = "0002-9890 (print), 1930-0972 (electronic)",
ISSN-L = "0002-9890",
bibdate = "Mon Jun 28 12:36:29 MDT 1999",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib; JSTOR
database",
note = "See erratum \cite{Smith:1896:EHS}.",
acknowledgement = ack-nhfb,
fjournal = "American Mathematical Monthly",
journal-URL = "https://www.jstor.org/journals/00029890.htm",
}
@Article{Smith:1896:EHS,
author = "D. E. Smith",
title = "Errata: Historical Survey of the Attempts at the
Computation and Construction of $ \pi $",
journal = j-AMER-MATH-MONTHLY,
volume = "3",
number = "2",
pages = "60--60",
month = feb,
year = "1896",
CODEN = "AMMYAE",
ISSN = "0002-9890 (print), 1930-0972 (electronic)",
ISSN-L = "0002-9890",
bibdate = "Mon Jun 28 12:36:34 MDT 1999",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib; JSTOR
database",
note = "See \cite{Smith:1895:HSA}.",
acknowledgement = ack-nhfb,
fjournal = "American Mathematical Monthly",
journal-URL = "https://www.jstor.org/journals/00029890.htm",
}
@Article{Veblen:1904:T,
author = "Oswald Veblen",
title = "The Transcendence of $ \pi $ and $e$",
journal = j-AMER-MATH-MONTHLY,
volume = "11",
number = "12",
pages = "219--223",
month = dec,
year = "1904",
CODEN = "AMMYAE",
ISSN = "0002-9890 (print), 1930-0972 (electronic)",
ISSN-L = "0002-9890",
bibdate = "Mon Jun 28 12:37:32 MDT 1999",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib; JSTOR
database",
acknowledgement = ack-nhfb,
fjournal = "American Mathematical Monthly",
journal-URL = "https://www.jstor.org/journals/00029890.htm",
}
@Article{Ramanujan:1914:MEA,
author = "Srinivasa Ramanujan",
title = "Modular equations and approximations to $ \pi $",
journal = "Quarterly Journal of Mathematics",
volume = "45",
number = "??",
pages = "180, 350--372",
month = "????",
year = "1914",
MRclass = "01A75",
MRnumber = "2280849",
bibdate = "Fri Jan 09 12:45:21 2015",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
note = "Reprinted in \cite[pages 23--39]{Hardy:1927:CPS} and
in \cite{Hardy:2000:CPS}.",
ZMnumber = "45.0688.02",
acknowledgement = ack-nhfb,
ajournal = "Quart. J. Math (or Q. J. Math.)",
fjournal = "Quarterly Journal of Mathematics",
}
@Article{Archibald:1921:HNR,
author = "R. C. Archibald",
title = "Historical Notes on the Relation $ e^{-(\pi / 2)} =
i^i $",
journal = j-AMER-MATH-MONTHLY,
volume = "28",
number = "3",
pages = "116--121",
month = mar,
year = "1921",
CODEN = "AMMYAE",
ISSN = "0002-9890 (print), 1930-0972 (electronic)",
ISSN-L = "0002-9890",
bibdate = "Mon Jun 28 12:36:09 MDT 1999",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib; JSTOR
database",
acknowledgement = ack-nhfb,
fjournal = "American Mathematical Monthly",
journal-URL = "https://www.jstor.org/journals/00029890.htm",
}
@Article{Underwood:1924:QDD,
author = "R. S. Underwood",
title = "Questions and Discussions: Discussions: Some Results
Involving $ \pi $",
journal = j-AMER-MATH-MONTHLY,
volume = "31",
number = "8",
pages = "392--394",
month = oct,
year = "1924",
CODEN = "AMMYAE",
ISSN = "0002-9890 (print), 1930-0972 (electronic)",
ISSN-L = "0002-9890",
bibdate = "Mon Jun 28 12:37:24 MDT 1999",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib; JSTOR
database",
acknowledgement = ack-nhfb,
fjournal = "American Mathematical Monthly",
journal-URL = "https://www.jstor.org/journals/00029890.htm",
}
@Article{Bennett:1925:QDT,
author = "A. A. Bennett",
title = "Questions and Discussions: Two New Arctangent
Relations for $ \pi $",
journal = j-AMER-MATH-MONTHLY,
volume = "32",
number = "5",
pages = "253--255",
month = may,
year = "1925",
CODEN = "AMMYAE",
ISSN = "0002-9890 (print), 1930-0972 (electronic)",
ISSN-L = "0002-9890",
bibdate = "Mon Jun 28 12:37:40 MDT 1999",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib; JSTOR
database",
acknowledgement = ack-nhfb,
fjournal = "American Mathematical Monthly",
journal-URL = "https://www.jstor.org/journals/00029890.htm",
}
@Article{Camp:1926:QDDb,
author = "C. C. Camp",
title = "Questions and Discussions: Discussions: a New
Calculation of $ \pi $",
journal = j-AMER-MATH-MONTHLY,
volume = "33",
number = "9",
pages = "472--473",
month = nov,
year = "1926",
CODEN = "AMMYAE",
DOI = "https://doi.org/10.2307/2299614",
ISSN = "0002-9890 (print), 1930-0972 (electronic)",
ISSN-L = "0002-9890",
MRnumber = "1521028",
bibdate = "Mon Jun 28 12:38:12 MDT 1999",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib; JSTOR
database",
acknowledgement = ack-nhfb,
fjournal = "American Mathematical Monthly",
journal-URL = "https://www.jstor.org/journals/00029890.htm",
}
@Article{Schoy:1926:QDDb,
author = "Carl Schoy",
title = "Questions and Discussions: Discussions: {Al-Biruni}'s
Computation of the Value of $ \pi $",
journal = j-AMER-MATH-MONTHLY,
volume = "33",
number = "6",
pages = "323--325",
month = jun # "\slash " # jul,
year = "1926",
CODEN = "AMMYAE",
ISSN = "0002-9890 (print), 1930-0972 (electronic)",
ISSN-L = "0002-9890",
bibdate = "Mon Jun 28 12:38:06 MDT 1999",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib; JSTOR
database",
acknowledgement = ack-nhfb,
fjournal = "American Mathematical Monthly",
journal-URL = "https://www.jstor.org/journals/00029890.htm",
}
@Book{Hardy:1927:CPS,
editor = "G. H. (Godfrey Harold) Hardy and P. V. (P.
Venkatesvara) {Seshu Aiyar} and B. M. (Bertram Martin)
Wilson",
title = "Collected papers of {Srinivasa Ramanujan}",
publisher = pub-CAMBRIDGE,
address = pub-CAMBRIDGE:adr,
pages = "xxxvi + 355 + 1",
year = "1927",
LCCN = "QA3 .R3",
bibdate = "Fri Jan 9 12:48:06 MST 2015",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib;
z3950.loc.gov:7090/Voyager",
acknowledgement = ack-nhfb,
author-dates = "1887--1920",
remark = "See entry \cite{Hardy:2000:CPS} for table of contents
of reprinted edition.",
subject = "Mathematics",
}
@Article{Ganguli:1930:EAV,
author = "Saradakanta Ganguli",
title = "The Elder {Aryabhata}'s Value of $ \pi $",
journal = j-AMER-MATH-MONTHLY,
volume = "37",
number = "1",
pages = "16--22",
month = jan,
year = "1930",
CODEN = "AMMYAE",
ISSN = "0002-9890 (print), 1930-0972 (electronic)",
ISSN-L = "0002-9890",
bibdate = "Mon Jun 28 12:35:44 MDT 1999",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib; JSTOR
database",
acknowledgement = ack-nhfb,
fjournal = "American Mathematical Monthly",
journal-URL = "https://www.jstor.org/journals/00029890.htm",
}
@Article{Lowry:1931:C,
author = "H. V. Lowry",
title = "The calculation of $ \pi $",
journal = j-MATH-GAZ,
volume = "15",
pages = "502--503",
year = "1931",
CODEN = "MAGAAS",
ISSN = "0025-5572",
bibdate = "Mon Apr 25 17:10:47 2011",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
ZMnumber = "57.0692.01",
abstract = "Verbesserung der aus der Betrachtung des $ 2^n$-Ecks
entspringenden Quadratwurzelmethode zur
n{\"a}herungsweisen Berechnung von $ \pi $. (V 3.).",
acknowledgement = ack-nhfb,
fjournal = "Mathematical Gazette",
journal-URL = "http://www.m-a.org.uk/jsp/index.jsp?lnk=620",
reviewer = "Wielandt, H.",
}
@Article{Barbour:1933:SCC,
author = "J. M. Barbour",
title = "A Sixteenth Century {Chinese} Approximation for $ \pi
$",
journal = j-AMER-MATH-MONTHLY,
volume = "40",
number = "2",
pages = "69--73",
month = feb,
year = "1933",
CODEN = "AMMYAE",
ISSN = "0002-9890 (print), 1930-0972 (electronic)",
ISSN-L = "0002-9890",
bibdate = "Mon Jun 28 12:36:54 MDT 1999",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib; JSTOR
database",
acknowledgement = ack-nhfb,
fjournal = "American Mathematical Monthly",
journal-URL = "https://www.jstor.org/journals/00029890.htm",
}
@Article{Frame:1935:QDN,
author = "J. S. Frame",
title = "Questions, Discussions, and Notes: a Series Useful in
the Computation of $ \pi $",
journal = j-AMER-MATH-MONTHLY,
volume = "42",
number = "8",
pages = "499--501",
month = oct,
year = "1935",
CODEN = "AMMYAE",
DOI = "https://doi.org/10.2307/2300475",
ISSN = "0002-9890 (print), 1930-0972 (electronic)",
ISSN-L = "0002-9890",
MRnumber = "1523462",
bibdate = "Mon Jun 28 12:37:55 MDT 1999",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib; JSTOR
database",
acknowledgement = ack-nhfb,
fjournal = "American Mathematical Monthly",
journal-URL = "https://www.jstor.org/journals/00029890.htm",
}
@Article{Uvanovic:1936:IPE,
author = "Daniel Uvanovi{\'c}",
title = "The {Indian} Prelude to {European} Mathematics",
journal = j-OSIRIS,
volume = "1",
number = "??",
pages = "652--657",
month = jan,
year = "1936",
CODEN = "OSIRAX",
DOI = "https://doi.org/10.2307/301630",
ISSN = "0369-7827 (print), 1933-8287 (electronic)",
ISSN-L = "0369-7827",
bibdate = "Mon Mar 30 15:08:54 MDT 2015",
bibsource = "http://www.jstor.org/action/showPublication?journalCode=osiris;
http://www.jstor.org/stable/i213312;
https://www.math.utah.edu/pub/tex/bib/osiris.bib;
https://www.math.utah.edu/pub/tex/bib/pi.bib",
URL = "http://www.jstor.org/stable/301630",
acknowledgement = ack-nhfb,
fjournal = "Osiris",
journal-URL = "http://www.jstor.org/page/journal/osiris/about.html",
remark = "From page 655, the author states, without literature
references, ``From Bhaskara's [ca. 1150 CE] value of $
\pi $ correct to four decimal places to the value given
in a South Indian compilation, the Sadratnamala (c.
1530 [CE]), which is correct to more than a dozen
places, there remains a gap which cannot be bridged
over at present. Nothing like this was available in
Europe before the days of the Bernoullis. But the
Chinese in the 13th century had reached a value of $
\pi $ midway between these two Indian determinations.
In that age imperial embassies from China to Ceylon and
to the South Indian kingdoms were not unknown. A
century later [ca. 1350 CE], an Indian manuscript notes
a value of $ \pi $ correct to about 30 decimal
places.''",
}
@Article{Gaba:1938:QDN,
author = "M. G. Gaba",
title = "Questions, Discussions, and Notes: a Simple
Approximation for $ \pi $",
journal = j-AMER-MATH-MONTHLY,
volume = "45",
number = "6",
pages = "373--375",
month = jun # "\slash " # jul,
year = "1938",
CODEN = "AMMYAE",
ISSN = "0002-9890 (print), 1930-0972 (electronic)",
ISSN-L = "0002-9890",
bibdate = "Mon Jun 28 12:38:57 MDT 1999",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib; JSTOR
database",
acknowledgement = ack-nhfb,
fjournal = "American Mathematical Monthly",
journal-URL = "https://www.jstor.org/journals/00029890.htm",
}
@Article{Lehmer:1938:AR,
author = "D. H. Lehmer",
title = "On Arccotangent Relations for $ \pi $",
journal = j-AMER-MATH-MONTHLY,
volume = "45",
number = "10",
pages = "657--664",
month = dec,
year = "1938",
CODEN = "AMMYAE",
ISSN = "0002-9890 (print), 1930-0972 (electronic)",
ISSN-L = "0002-9890",
bibdate = "Mon Jun 28 12:39:07 MDT 1999",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib; JSTOR
database",
acknowledgement = ack-nhfb,
fjournal = "American Mathematical Monthly",
journal-URL = "https://www.jstor.org/journals/00029890.htm",
}
@Article{Ballantine:1939:QDNb,
author = "J. P. Ballantine",
title = "Questions, Discussions, and Notes: The Best (?)
Formula for Computing $ \pi $ to a Thousand Places",
journal = j-AMER-MATH-MONTHLY,
volume = "46",
number = "8",
pages = "499--501",
month = oct,
year = "1939",
CODEN = "AMMYAE",
ISSN = "0002-9890 (print), 1930-0972 (electronic)",
ISSN-L = "0002-9890",
bibdate = "Mon Jun 28 12:39:26 MDT 1999",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib; JSTOR
database",
acknowledgement = ack-nhfb,
fjournal = "American Mathematical Monthly",
journal-URL = "https://www.jstor.org/journals/00029890.htm",
}
@Article{Niven:1939:T,
author = "Ivan Niven",
title = "The Transcendence of $ \pi $",
journal = j-AMER-MATH-MONTHLY,
volume = "46",
number = "8",
pages = "469--471",
month = oct,
year = "1939",
CODEN = "AMMYAE",
ISSN = "0002-9890 (print), 1930-0972 (electronic)",
ISSN-L = "0002-9890",
bibdate = "Mon Jun 28 12:39:26 MDT 1999",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib; JSTOR
database",
acknowledgement = ack-nhfb,
fjournal = "American Mathematical Monthly",
journal-URL = "https://www.jstor.org/journals/00029890.htm",
}
@Article{Thomas:1940:RPZ,
author = "G. B. Thomas",
title = "Recent Publications: {{\em Die Zahl $ \pi $ der
Kreis}}, by {Franz Hennecke}",
journal = j-AMER-MATH-MONTHLY,
volume = "47",
number = "8",
pages = "560--561",
month = oct,
year = "1940",
CODEN = "AMMYAE",
ISSN = "0002-9890 (print), 1930-0972 (electronic)",
ISSN-L = "0002-9890",
bibdate = "Mon Jun 28 12:36:00 MDT 1999",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib; JSTOR
database",
acknowledgement = ack-nhfb,
fjournal = "American Mathematical Monthly",
journal-URL = "https://www.jstor.org/journals/00029890.htm",
}
@Article{Dorwart:1942:DNV,
author = "H. L. Dorwart",
title = "Discussions and Notes: Values of the Trigonometric
Ratios of $ \pi / 8 $ and $ \pi / 12 $",
journal = j-AMER-MATH-MONTHLY,
volume = "49",
number = "5",
pages = "324--325",
month = may,
year = "1942",
CODEN = "AMMYAE",
ISSN = "0002-9890 (print), 1930-0972 (electronic)",
ISSN-L = "0002-9890",
bibdate = "Mon Jun 28 12:36:39 MDT 1999",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib; JSTOR
database",
acknowledgement = ack-nhfb,
fjournal = "American Mathematical Monthly",
journal-URL = "https://www.jstor.org/journals/00029890.htm",
}
@Article{Menger:1945:MP,
author = "Karl Menger",
title = "Methods of Presenting $e$ and $ \pi $",
journal = j-AMER-MATH-MONTHLY,
volume = "52",
number = "1",
pages = "28--33",
month = jan,
year = "1945",
CODEN = "AMMYAE",
ISSN = "0002-9890 (print), 1930-0972 (electronic)",
ISSN-L = "0002-9890",
bibdate = "Mon Jun 28 12:37:38 MDT 1999",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib; JSTOR
database",
acknowledgement = ack-nhfb,
fjournal = "American Mathematical Monthly",
journal-URL = "https://www.jstor.org/journals/00029890.htm",
}
@Article{Copeland:1946:NNN,
author = "Arthur H. Copeland and Paul Erd{\H{o}}s",
title = "Note on normal numbers",
journal = j-BULL-AMS,
volume = "52",
pages = "857--860",
year = "1946",
CODEN = "BAMOAD",
ISSN = "0002-9904 (print), 1936-881X (electronic)",
ISSN-L = "0002-9904",
MRclass = "10.0X",
MRnumber = "0017743 (8,194b)",
MRreviewer = "R. D. James",
bibdate = "Fri May 3 18:38:50 2013",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
acknowledgement = ack-nhfb,
fjournal = "Bulletin of the American Mathematical Society",
keywords = "Champernowne normal decimal numbers",
remark-1 = "See \cite[page 377]{Bailey:2012:EAN} for the
significance of this work.",
remark-2 = "This paper generalizes Champernowne's construction of
specific normal decimal numbers.",
}
@Article{Ferguson:1946:EPS,
author = "D. F. Ferguson",
title = "Evaluation of pi: Are {Shanks}' Figures Correct?",
journal = j-MATH-GAZ,
volume = "30",
number = "289",
pages = "89--90",
month = may,
year = "1946",
CODEN = "MAGAAS",
ISSN = "0025-5572",
bibdate = "Fri Jul 01 06:42:18 2011",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
URL = "http://www.jstor.org/stable/3608485",
acknowledgement = ack-nhfb,
fjournal = "Mathematical Gazette",
journal-URL = "http://www.m-a.org.uk/jsp/index.jsp?lnk=620",
remark = "Ferguson uses the series $ \pi / 4 = 3 \tan^{-1}(1 /
4) + \tan^{-1}(1 / 20) + \tan^{-1}(1 / 1985) $,
credited to his colleague R. W. Morris, and finds
disagreement at the 530th decimal place with Shanks
results of 1853 and 1873. He comments at the bottom of
the first page ``I give the figures from the 521st
place to the 540th place (i) as Shanks gave them, (ii)
as I think they should be: (i) 86021 39501 60924 48077
(Shanks), (ii) 86021 39494 63952 24737 (D. F. F.).''. A
modern calculation in Maple with evalf(Pi,561) produces
the last 40 digits as 86021 39494 63952 24737 19070
21798 60943 70277 \ldots{}. Thus, Ferguson's
conclusion, and his results, are correct. Ferguson
describes his hand calculation as taking about one
year. The Maple computation takes a few milliseconds
(less than the timer tick size).",
}
@Article{Anonymous:1947:NA,
author = "Anonymous",
title = "A New Approximation to $ \pi $",
journal = j-MATH-TABLES-OTHER-AIDS-COMPUT,
volume = "2",
number = "18",
pages = "245--248",
month = apr,
year = "1947",
CODEN = "MTTCAS",
ISSN = "0891-6837",
bibdate = "Tue Oct 13 08:44:19 MDT 1998",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib; JSTOR
database",
acknowledgement = ack-nhfb,
fjournal = "Mathematical Tables and Other Aids to Computation",
journal-URL = "http://www.ams.org/mcom/",
}
@Article{Smith:1947:NA,
author = "L. B. Smith and J. W. Wrench and D. F. Ferguson",
title = "A New Approximation to $ \pi $",
journal = j-MATH-TABLES-OTHER-AIDS-COMPUT,
volume = "2",
number = "18",
pages = "245--248",
month = apr,
year = "1947",
CODEN = "MTTCAS",
ISSN = "0891-6837",
bibdate = "Fri Jul 01 09:03:49 2011",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib; JSTOR
database",
URL = "http://www.jstor.org/stable/2002296",
acknowledgement = ack-nhfb,
fjournal = "Mathematical Tables and Other Aids to Computation",
journal-URL = "http://www.ams.org/mcom/",
remark = "The authors use the expansion of $ p i / 4 $ in arc
tangent terms to obtain about 800 digits of $ \pi $.
See \cite{Ferguson:1948:NAC} for confirmation to 812
digits.",
}
@Article{Ferguson:1948:NAC,
author = "D. F. Ferguson and John W. {Wrench, Jr.}",
title = "A New Approximation to $ \pi $ (Conclusion)",
journal = j-MATH-TABLES-OTHER-AIDS-COMPUT,
volume = "3",
number = "21",
pages = "18--19",
month = jan,
year = "1948",
CODEN = "MTTCAS",
ISSN = "0891-6837",
bibdate = "Tue Oct 13 08:44:19 MDT 1998",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib; JSTOR
database",
URL = "http://www.jstor.org/stable/2002657",
acknowledgement = ack-nhfb,
fjournal = "Mathematical Tables and Other Aids to Computation",
journal-URL = "http://www.ams.org/mcom/",
remark = "The authors report an error in previous work, and
produce these digits of $ \pi $ for the interval
721D--808D: 86403 44181 59813 62977 47713 09960 51870
72113 49999 99837 29780 49951 05973 17328 16096 31859
50244 594(55). A modern computation in Maple with
evalf(Pi, 823) produces the digits 86403 44181 59813
62977 47713 09960 51870 72113 49999 99837 29780 49951
05973 17328 16096 31859 50244 59455 34690 83026
\ldots{}, confirming the last 5 computed digits of $
\pi $ this paper. This result of 808 decimal digits may
have been the last published hand calculation of digits
of $ \pi $, after which computers were used to rapidly
advance the known digits.",
}
@InProceedings{Eisenhart:1950:RDD,
author = "Eisenhart and L. S. Deming",
booktitle = "{National Bureau of Standards Seminar, February 17,
Washington, DC}",
title = "On the randomness of the digits of $ \pi $ and $e$ to
2000 decimal places",
publisher = "????",
address = "????",
pages = "??--??",
year = "1950",
bibdate = "Mon Jan 16 14:24:10 2012",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib;
https://www.math.utah.edu/pub/tex/bib/prng.bib",
acknowledgement = ack-nhfb,
}
@Article{Metropolis:1950:STV,
author = "N. C. Metropolis and G. Reitwiesner and J. von
Neumann",
title = "Statistical treatment of values of first $ 2, 000 $
decimal digits of {$e$} and of {$ \pi $} calculated on
the {ENIAC}",
journal = j-MATH-TABLES-OTHER-AIDS-COMPUT,
volume = "4",
number = "30",
pages = "109--111",
year = "1950",
CODEN = "MTTCAS",
ISSN = "0891-6837",
MRclass = "65.0X",
MRnumber = "MR0037598 (12,286j)",
MRreviewer = "R. P. Boas, Jr.",
bibdate = "Mon Jun 06 19:17:03 2005",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib;
MathSciNet database",
abstract = "From the article: ``The first 2,000 decimal digits of
$e$ and $ \pi $ were calculated on the ENIAC by Mr. G.
Reitwiesner and several members of the ENIAC Branch of
the Ballistic Research Laboratories at Aberdeen,
Maryland \cite{Reitwiesner:1950:EDM}. A statistical
survey of this material has failed to disclose an
significant deviations from randomness for $ \pi $, but
it has indicated quite serious ones for $e$.''",
acknowledgement = ack-nhfb,
fjournal = "Mathematical Tables and Other Aids to Computation",
journal-URL = "http://www.ams.org/mcom/",
}
@Article{Reitwiesner:1950:EDM,
author = "George W. Reitwiesner",
title = "An {ENIAC} Determination of $ \pi $ and $e$ to more
than 2000 Decimal Places",
journal = j-MATH-TABLES-OTHER-AIDS-COMPUT,
volume = "4",
number = "29",
pages = "11--15",
month = jan,
year = "1950",
CODEN = "MTTCAS",
ISSN = "0891-6837",
MRclass = "65.0X",
MRnumber = "0037597 (12,286i)",
MRreviewer = "R. P. Boas, Jr.",
bibdate = "Tue Oct 13 08:06:19 MDT 1998",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib; JSTOR
database",
URL = "http://www.jstor.org/stable/2002695",
acknowledgement = ack-nhfb,
fjournal = "Mathematical Tables and Other Aids to Computation",
journal-URL = "http://www.ams.org/mcom/",
remark = "This paper reports 2035 digits of $ \pi $ nd 2010
digits of $e$. The computation took 11 hours for $e$
and $ 70$ hours for $ \pi $, including machine time and
punched-card-handling time.",
}
@Article{Schepler:1950:CPa,
author = "Herman C. Schepler",
title = "The Chronology of Pi",
journal = j-MATH-MAG,
volume = "23",
number = "3",
pages = "165--170",
month = jan # "\slash " # feb,
year = "1950",
CODEN = "MAMGA8",
DOI = "https://doi.org/10.2307/3029284",
ISSN = "0025-570X",
ISSN-L = "0025-570X",
bibdate = "Wed Oct 21 10:38:44 2015",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
URL = "http://www.jstor.org/stable/3029284",
acknowledgement = ack-nhfb,
fjournal = "Mathematics Magazine",
journal-URL = "http://www.maa.org/pubs/mathmag.html",
remark = "Covers pi computation from 3000 BCE to 628 CE.",
}
@Article{Schepler:1950:CPb,
author = "Herman C. Schepler",
title = "The Chronology of Pi",
journal = j-MATH-MAG,
volume = "23",
number = "4",
pages = "216--228",
month = mar # "\slash " # apr,
year = "1950",
CODEN = "MAMGA8",
DOI = "https://doi.org/10.2307/3029832",
ISSN = "0025-570X",
ISSN-L = "0025-570X",
bibdate = "Wed Oct 21 10:38:44 2015",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
URL = "http://www.jstor.org/stable/3029832",
acknowledgement = ack-nhfb,
fjournal = "Mathematics Magazine",
journal-URL = "http://www.maa.org/pubs/mathmag.html",
remark = "Covers pi computation from 825 to 1868.",
}
@Article{Schepler:1950:CPc,
author = "Herman C. Schepler",
title = "The Chronology of Pi",
journal = j-MATH-MAG,
volume = "23",
number = "5",
pages = "279--283",
month = may # "\slash " # jun,
year = "1950",
CODEN = "MAMGA8",
DOI = "https://doi.org/10.2307/3029000",
ISSN = "0025-570X",
ISSN-L = "0025-570X",
bibdate = "Wed Oct 21 10:38:44 2015",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
URL = "http://www.jstor.org/stable/3029000",
acknowledgement = ack-nhfb,
fjournal = "Mathematics Magazine",
journal-URL = "http://www.maa.org/pubs/mathmag.html",
remark = "Covers pi computation from 1872 to 1949. The last
entry is ``1949. U.S. Army (U.S.A.)./2,035 places.
Yielding to an irresistible temptation, some
mathematical machine operators presented the problem of
evaluating $ \pi $ to Eniac, the all-electronic
calculator at the Army's Ballistic Research
Laboratories in Aberdeen, Maryland. The machine's
18,800 electron tubes went into action and computed $
\pi $ to 2,035 places in about 70 hours. In 1873,
William Shanks gave the value of $ \pi $ to 707 decimal
places (527 correct). The computation took him more
than 15 years. Scientific American, Dec., 1949, p. 30
and Feb., 1950, p. 2.''",
}
@Article{Breusch:1954:MNP,
author = "Robert Breusch",
title = "Mathematical Notes: a Proof of the Irrationality of $
\pi $",
journal = j-AMER-MATH-MONTHLY,
volume = "61",
number = "9",
pages = "631--632",
month = nov,
year = "1954",
CODEN = "AMMYAE",
ISSN = "0002-9890 (print), 1930-0972 (electronic)",
ISSN-L = "0002-9890",
bibdate = "Mon Jun 28 12:37:38 MDT 1999",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib; JSTOR
database",
acknowledgement = ack-nhfb,
fjournal = "American Mathematical Monthly",
journal-URL = "https://www.jstor.org/journals/00029890.htm",
}
@Article{Greenwood:1955:CCT,
author = "Robert E. Greenwood",
title = "Coupon Collector's Test for Random Digits",
journal = j-MATH-TABLES-OTHER-AIDS-COMPUT,
volume = "9",
number = "49",
pages = "1--5",
month = jan,
year = "1955",
CODEN = "MTTCAS",
ISSN = "0891-6837",
bibdate = "Tue Oct 13 08:06:19 MDT 1998",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib; JSTOR
database",
URL = "http://www.jstor.org/stable/2002211",
abstract = "Increasing use of random numbers, especially in Monte
Carlo procedures and in large computing installations,
has served to focus attention on the various tests for
randomness. Kendall and Babington-Smith list four tests
for so-called local randomness. While not giving the
coupon collector's test (to be described below) a place
in their now classical list of four tests, they did use
a modified coupon collector's test in some of their
investigations.",
acknowledgement = ack-nhfb,
fjournal = "Mathematical Tables and Other Aids to Computation",
journal-URL = "http://www.ams.org/mcom/",
remark = "This paper discusses chi-square tests for randomness
on the decimal digits of $ \pi $ and $e$. A 2035-digit
value of $ \pi $ \cite{Reitwiesner:1950:EDM}, a
2010-digit value of $e$ \cite{Reitwiesner:1950:EDM},
and a 2500-digit value of $e$
\cite{Metropolis:1950:STV}, were used in the tests, and
the author concludes with ``Neither of these chi-square
test values is unusually out of line.''.",
}
@Article{Kazarinoff:1955:CNS,
author = "D. K. Kazarinoff",
title = "Classroom Notes: a Simple Derivation of the
{Leibnitz-Gregory} Series for $ \pi / 4 $",
journal = j-AMER-MATH-MONTHLY,
volume = "62",
number = "10",
pages = "726--727",
month = dec,
year = "1955",
CODEN = "AMMYAE",
ISSN = "0002-9890 (print), 1930-0972 (electronic)",
ISSN-L = "0002-9890",
bibdate = "Mon Jun 28 12:38:04 MDT 1999",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib; JSTOR
database",
acknowledgement = ack-nhfb,
fjournal = "American Mathematical Monthly",
journal-URL = "https://www.jstor.org/journals/00029890.htm",
}
@Article{Nicholson:1955:SCN,
author = "S. C. Nicholson and J. Jeenel",
title = "Some Comments on a {NORC} Computation of $ \pi $",
journal = j-MATH-TABLES-OTHER-AIDS-COMPUT,
volume = "9",
number = "52",
pages = "162--164",
month = oct,
year = "1955",
CODEN = "MTTCAS",
ISSN = "0891-6837",
MRclass = "65.0X",
MRnumber = "0075672 (17,789b)",
MRreviewer = "D. H. Lehmer",
bibdate = "Tue Oct 13 08:06:19 MDT 1998",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib; JSTOR
database; MathSciNet database",
URL = "http://www.jstor.org/stable/2002052",
acknowledgement = ack-nhfb,
fjournal = "Mathematical Tables and Other Aids to Computation",
journal-URL = "http://www.ams.org/mcom/",
remark = "The paper reports 3089 digits of $ \pi $ obtained in
13 minutes of computation. It also observes: ``if the
time to compute $ \pi $ to $m$ digits is $t$ units,
then the time to produce $ k m$ digits is roughly $ k^2
t$ units; this holds true as long as the calculation is
contained in high-speed storage.''",
}
@Article{Pennisi:1955:CNE,
author = "L. L. Pennisi",
title = "Classroom Notes: Expansions for $ \pi $ and $ \pi^2
$",
journal = j-AMER-MATH-MONTHLY,
volume = "62",
number = "9",
pages = "653--654",
month = nov,
year = "1955",
CODEN = "AMMYAE",
ISSN = "0002-9890 (print), 1930-0972 (electronic)",
ISSN-L = "0002-9890",
bibdate = "Mon Jun 28 12:38:02 MDT 1999",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib; JSTOR
database",
acknowledgement = ack-nhfb,
fjournal = "American Mathematical Monthly",
journal-URL = "https://www.jstor.org/journals/00029890.htm",
}
@InProceedings{Felton:1957:ECM,
author = "G. E. Felton",
editor = "Anonymous",
booktitle = "{Abbreviated proceedings of the Oxford Mathematical
Conference for Schoolteachers and Industrialists at
Trinity College, Oxford, April 8--18, 1957 and
administered by Oxford University Delegacy for
Extra-Mural Studies}",
title = "Electronic Computers and Mathematicians",
publisher = "Technology (The Times Publishing Company Limited)",
address = "London, UK",
bookpages = "111",
pages = "12--17",
year = "1957",
LCCN = "QA11.A1 O9 1957",
bibdate = "Fri Jul 1 09:32:16 MDT 2011",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib;
library.ox.ac.uk:210/ADVANCE",
note = "Footnote 12-53.",
acknowledgement = ack-nhfb,
remark = "Felton reports 10,000 digits of $ \pi $ obtained in 33
hours on the Pegasus computer at the Ferranti Computer
Center in London, using Klingenstierna's (1730)
relation $ \pi / 4 = 8 \arctan (1 / 10) - \arctan (1 /
239) - 4 \arctan (1 / 515) $. The formula was
rediscovered by Schellbach in 1832. Due to a machine
error, Felton's result is only correct to 7480 decimal
places.",
}
@InBook{Steinhaus:1958:PCB,
author = "H. Steinhaus",
booktitle = "The New {Scottish} Book, 1946--1958",
title = "Problem 144: [conjecture on base-dependence of normal
numbers]",
publisher = "????",
address = "Wroc{\l}aw, Poland",
bookpages = "????",
year = "1958",
LCCN = "????",
bibdate = "Sat Jan 07 16:58:57 2012",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
note = "See \cite{Cassels:1959:PSA} for a negative answer to
this conjecture.",
acknowledgement = ack-nhfb,
remark = "I cannot find this book in major online catalogs, or
in the MathSciNet database, or in the ZMath database.",
}
@Article{Cassels:1959:PSA,
author = "J. W. S. Cassels",
title = "On a problem of {Steinhaus} about normal numbers",
journal = j-COLLOQ-MATH,
volume = "7",
pages = "95--101",
year = "1959",
CODEN = "CQMAAQ",
ISSN = "0010-1354 (print), 1730-6302 (electronic)",
ISSN-L = "0010-1354",
MRclass = "10.00",
MRnumber = "0113863 (22 \#4694)",
MRreviewer = "N. G. de Bruijn",
bibdate = "Sat Jan 7 16:55:17 2012",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
note = "See \cite{Steinhaus:1958:PCB} for the original
problem.",
URL = "http://matwbn.icm.edu.pl/ksiazki/cm/cm7/cm7120.pdf",
acknowledgement = ack-nhfb,
fjournal = "Colloquium Mathematicum",
}
@Article{Schmidt:1960:NN,
author = "Wolfgang M. Schmidt",
title = "On normal numbers",
journal = j-PAC-J-MATH,
volume = "10",
pages = "661--672",
year = "1960",
CODEN = "PJMAAI",
ISSN = "0030-8730 (print), 1945-5844 (electronic)",
ISSN-L = "0030-8730",
MRclass = "10.00",
MRnumber = "0117212 (22 \#7994)",
MRreviewer = "F. Herzog",
bibdate = "Sat Jan 7 16:44:42 2012",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
URL = "http://projecteuclid.org/euclid.pjm/1103038420",
ZMnumber = "0093.05401",
acknowledgement = ack-nhfb,
fjournal = "Pacific Journal of Mathematics",
remark = "From the first section of the paper: ``In this paper
we solve the following problem. {\em Under what
conditions on $r$, $s$ is every number $ \xi $ which is
normal to base $r$ also normal to base $s$ ?} The
answer is given by: THEOREM 1. {\bf A} Assume $ r \sim
s$. Then any number normal to base $r$ is normal to
base $s$. {\bf B} If $ r \not \sim s$, then the set of
numbers $ \xi $ which are normal to base $r$ but not
even simply normal to base $s$ has the power of the
continuum.'' Here, the relation $ r \sim s$ means that
the exist integer $m$ and $n$ such that $ r^m = s^n$.",
}
@Article{Wrench:1960:EED,
author = "J. W. {Wrench, Jr.}",
title = "The Evolution of Extended Decimal Approximation to $
\pi $",
journal = j-MATH-TEACH,
volume = "53",
number = "??",
pages = "644--650",
month = dec,
year = "1960",
ISSN = "0025-5769",
bibdate = "Fri Jul 01 10:19:45 2011",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
note = "Reprinted in \cite[pp. 319--325]{Berggren:1997:PSB}.",
acknowledgement = ack-nhfb,
fjournal = "The Mathematics Teacher",
remark = "The author reports chi-square tests on the first 16167
decimal digits of $ \pi $, and finds no abnormal
behavior.",
xxnote = "The publisher Web site at
http://www.nctm.org/eresources/archive.asp?journal_id=2
has journal content only back to February 1997 (volume
90, number 2). The journal is not in the JSTOR
archive.",
}
@Article{Matsuoka:1961:MNE,
author = "Yoshio Matsuoka",
title = "Mathematical Notes: An Elementary Proof of the Formula
$ {\sum }^\infty_{k = 1} 1 / k^2 = \pi^2 / 6 $",
journal = j-AMER-MATH-MONTHLY,
volume = "68",
number = "5",
pages = "485--487",
month = may,
year = "1961",
CODEN = "AMMYAE",
ISSN = "0002-9890 (print), 1930-0972 (electronic)",
ISSN-L = "0002-9890",
bibdate = "Mon Jun 28 12:36:19 MDT 1999",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib; JSTOR
database",
acknowledgement = ack-nhfb,
fjournal = "American Mathematical Monthly",
journal-URL = "https://www.jstor.org/journals/00029890.htm",
}
@Article{Dixon:1962:MNA,
author = "J. D. Dixon",
title = "Mathematical Notes: $ \pi $ is not Algebraic of Degree
One or Two",
journal = j-AMER-MATH-MONTHLY,
volume = "69",
number = "7",
pages = "636--636",
month = aug # "\slash " # sep,
year = "1962",
CODEN = "AMMYAE",
ISSN = "0002-9890 (print), 1930-0972 (electronic)",
ISSN-L = "0002-9890",
bibdate = "Mon Jun 28 12:36:48 MDT 1999",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib; JSTOR
database",
acknowledgement = ack-nhfb,
fjournal = "American Mathematical Monthly",
journal-URL = "https://www.jstor.org/journals/00029890.htm",
}
@Book{Hardy:1962:CPS,
editor = "G. H. (Godfrey Harold) Hardy and P. V. (P.
Venkatesvara) {Seshu Aiyar} and B. M. (Bertram Martin)
Wilson",
title = "Collected papers of {Srinivasa Ramanujan}",
publisher = "Chelsea",
address = "New York, NY, USA",
pages = "vii + 355",
year = "1962",
LCCN = "QA3 .R3",
bibdate = "Fri Jan 9 12:48:06 MST 2015",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib;
z3950.loc.gov:7090/Voyager",
acknowledgement = ack-nhfb,
author-dates = "1887--1920",
remark = "See entry \cite{Hardy:2000:CPS} for table of contents
of reprinted edition.",
subject = "Mathematics",
}
@Article{Pathria:1962:SSR,
author = "R. K. Pathria",
title = "A Statistical Study of Randomness Among the First $
10, 000 $ Digits of $ \pi $",
journal = j-MATH-COMPUT,
volume = "16",
number = "78",
pages = "188--197",
month = apr,
year = "1962",
CODEN = "MCMPAF",
ISSN = "0025-5718 (print), 1088-6842 (electronic)",
ISSN-L = "0025-5718",
bibdate = "Tue Oct 13 08:06:19 MDT 1998",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib; JSTOR
database",
URL = "http://www.jstor.org/stable/2003057",
acknowledgement = ack-nhfb,
fjournal = "Mathematics of Computation",
journal-URL = "http://www.ams.org/mcom/",
}
@Article{Shanks:1962:CD,
author = "Daniel Shanks and John W. {Wrench, Jr.}",
title = "Calculation of $ \pi $ to 100,000 Decimals",
journal = j-MATH-COMPUT,
volume = "16",
number = "77",
pages = "76--99",
month = jan,
year = "1962",
CODEN = "MCMPAF",
ISSN = "0025-5718 (print), 1088-6842 (electronic)",
ISSN-L = "0025-5718",
MRclass = "65.99",
MRnumber = "0136051 (24 \#B2090)",
MRreviewer = "D. H. Lehmer",
bibdate = "Tue Oct 13 08:06:19 MDT 1998",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib; JSTOR
database; MathSciNet database",
note = "A note added in proof says: ``J. M. Gerard of IBM
United Kingdom Limited, who was then unaware of the
computation described above, computed $ \pi $ to 20,000
D on the 7090 in the London Data Centre on July 31,
1961. His program used Machin's formula, (1) [$ \pi =
16 \arctan (1 / 5) - 4 \arctan (1 / 239)$], and
required 39 minutes running time. His result agrees
with ours to that number of decimals.''",
URL = "http://www.jstor.org/stable/2003813",
acknowledgement = ack-nhfb,
fjournal = "Mathematics of Computation",
journal-URL = "http://www.ams.org/mcom/",
remark = "The computation required 8 hours 43 minutes on an IBM
7090 using St{\"o}rmer's (1896) formula, $ \pi = 24
\arctan (1 / 8) + 8 \arctan (1 / 57) + 4 \arctan (1 /
239) $.",
}
@Article{Smith:1966:CP,
author = "John Smith",
title = "The Challenge of {Pi}",
journal = j-IEEE-SPECTRUM,
volume = "3",
number = "10",
pages = "5--5",
month = oct,
year = "1966",
CODEN = "IEESAM",
DOI = "https://doi.org/10.1109/MSPEC.1966.5217340",
ISSN = "0018-9235 (print), 1939-9340 (electronic)",
ISSN-L = "0018-9235",
bibdate = "Wed Jan 15 08:45:04 2020",
bibsource = "https://www.math.utah.edu/pub/tex/bib/ieeespectrum1960.bib;
https://www.math.utah.edu/pub/tex/bib/pi.bib",
acknowledgement = ack-nhfb,
fjournal = "IEEE Spectrum",
journal-URL = "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=6",
keywords = "Computer errors; Digital arithmetic; Educational
institutions; Hardware; Physics computing; Power
engineering computing; Programming profession;
Registers; Testing; Upper bound",
remark = "Report of a computation of $ \pi $ to 17,935 places
using base-1,000,000 arithmetic. A footnote reports
``The AIL result to 17,940 places was identical with
the reference to 17,935 places. Reference: D. Shanks
and J. W. Wrench, Jr., `Calculation of Pi to 100,000
Decimals', Mathematics of Computation, January 1962,
Vol. 16, No. 77, pp. 67--99.''",
}
@Article{Esmenjaud-Bonnardel:1965:ESD,
author = "M. Esmenjaud-Bonnardel",
title = "{{\'E}}tude statistique des d{\'e}cimales de pi.
({French}) [{Statistical} study of the decimals of
pi]",
journal = j-CHIFFRES,
volume = "8",
number = "??",
pages = "295--306",
month = "????",
year = "1965",
CODEN = "????",
ISSN = "0245-9922",
bibdate = "Fri Jul 01 10:32:48 2011",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
acknowledgement = ack-nhfb,
fjournal = "Chiffres: Revue de l'Association fran{\c{c}}aise de
Calcul",
language = "French",
remark = "The author reports the results of four statistical
tests on the first 100,000 digits of $ \pi $
\cite{Shanks:1962:CD} and the first 100,000 digits of
the RAND million-random-digit corpus
\cite{RAND:1955:MRD}, and concludes that both are
random sequences.",
}
@Article{Good:1967:GST,
author = "I. J. Good and T. N. Gover",
title = "The generalized serial test and the binary expansion
of $ \sqrt {2} $",
journal = j-J-R-STAT-SOC-SER-A-GENERAL,
volume = "130",
number = "1",
pages = "102--107",
month = "????",
year = "1967",
CODEN = "JSSAEF",
ISSN = "0035-9238",
bibdate = "Sat Jan 07 11:23:58 2012",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib;
https://www.math.utah.edu/pub/tex/bib/prng.bib",
note = "See remark \cite{Good:1968:GST}.",
URL = "http://www.jstor.org/stable/2344040",
acknowledgement = ack-nhfb,
fjournal = "Journal of the Royal Statistical Society. Series A
(General)",
remark = "The first author reports in \cite[page 43, column
2]{Good:1969:HRR}: ``For binary sequences one of the
best tests is the generalized serial test. This test,
which uses a statistic having the appearance of a
`Chi-squared', is also useful when $ t \neq 2 $, but it
does not have asymptotically a chi-squared
distribution, a fact that has led to error in at least
five published papers
\cite{Forsythe:1951:GTRa,Kendall:1938:RRS,Pathria:1962:SSR,RAND:1955:MRD,Stoneham:1965:SDT}.
It would have led to the rejection of RAND's million
random digits if the test had been applied to many
blocks incorrectly, instead of to only a few. The
simple correct method of use is described in
\cite{Good:1967:GST} [this paper].''",
remark-2 = "Brief mention of the question of the normality of
$\pi$.",
}
@Article{Tee:1967:CP,
author = "G. J. Tee",
title = "Correspondence: $ \pi $ and pi",
journal = j-COMP-J,
volume = "9",
number = "4",
pages = "393--393",
month = feb,
year = "1967",
CODEN = "CMPJA6",
DOI = "https://doi.org/10.1093/comjnl/9.4.393",
ISSN = "0010-4620 (print), 1460-2067 (electronic)",
ISSN-L = "0010-4620",
bibdate = "Tue Dec 4 14:47:37 MST 2012",
bibsource = "http://comjnl.oxfordjournals.org/content/9/4.toc;
https://www.math.utah.edu/pub/tex/bib/compj2010.bib;
https://www.math.utah.edu/pub/tex/bib/pi.bib",
URL = "http://comjnl.oxfordjournals.org/content/9/4/393.full.pdf+html",
acknowledgement = ack-nhfb,
fjournal = "Computer Journal",
journal-URL = "http://comjnl.oxfordjournals.org/",
remark = "In this short letter, the author proposes generating
an explicit value of pi from an assignment of the
expression $ 4 \times \arctan (1) $. Similar ideas have
been rediscovered and repeated many times since, but
are almost always a bad idea because they rely on the
sometimes dubious accuracy of library routines over
which the programmer has little control, and expression
from which they are computed may introduce additional
rounding error (multiplication by 4 in a decimal or
octal or hexadecimal base in general requires one
rounding).",
}
@Article{Yarbrough:1967:PCC,
author = "Lynn Yarbrough",
title = "Precision calculations of $e$ and $ \pi $ constants",
journal = j-CACM,
volume = "10",
number = "9",
pages = "537--537",
month = sep,
year = "1967",
CODEN = "CACMA2",
ISSN = "0001-0782 (print), 1557-7317 (electronic)",
ISSN-L = "0001-0782",
bibdate = "Fri Nov 25 18:20:15 MST 2005",
bibsource = "http://www.acm.org/pubs/contents/journals/cacm/;
https://www.math.utah.edu/pub/tex/bib/cacm1960.bib;
https://www.math.utah.edu/pub/tex/bib/pi.bib",
acknowledgement = ack-nhfb,
fjournal = "Communications of the ACM",
journal-URL = "http://portal.acm.org/browse_dl.cfm?idx=J79",
keywords = "floating-point arithmetic; number base conversion",
remark = "Gives decimal, octal, and hexadecimal values of $e$
and $ \pi $ to 100 digits, and notes ``The difficulty
arises because assemblers and compilers are hardly ever
designed to convert decimal constants to a precision of
more than a dozen or so digits. Thus, if calculations
to greater precision are to be done, constants usually
must be input in octal or other binary-derived
representation.''.",
}
@Article{Good:1968:GST,
author = "I. J. Good and T. N. Gover",
title = "The generalized serial test and the binary expansion
of $ \sqrt {2} $",
journal = j-J-R-STAT-SOC-SER-A-GENERAL,
volume = "131",
number = "??",
pages = "434--434",
month = "????",
year = "1968",
CODEN = "JSSAEF",
ISSN = "0035-9238",
bibdate = "Sat Jan 07 11:23:58 2012",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib;
https://www.math.utah.edu/pub/tex/bib/prng.bib",
note = "See \cite{Good:1967:GST}.",
acknowledgement = ack-nhfb,
fjournal = "Journal of the Royal Statistical Society. Series A
(General)",
}
@Article{Brown:1969:REE,
author = "W. S. Brown",
title = "Rational Exponential Expressions and a Conjecture
Concerning $ \pi $ and $e$",
journal = j-AMER-MATH-MONTHLY,
volume = "76",
number = "1",
pages = "28--34",
month = jan,
year = "1969",
CODEN = "AMMYAE",
ISSN = "0002-9890 (print), 1930-0972 (electronic)",
ISSN-L = "0002-9890",
bibdate = "Mon Jun 28 12:39:15 MDT 1999",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib; JSTOR
database",
acknowledgement = ack-nhfb,
fjournal = "American Mathematical Monthly",
journal-URL = "https://www.jstor.org/journals/00029890.htm",
}
@Article{Draim:1969:FCF,
author = "N. A. Draim",
title = "$ \pi $ in the Form of a Continued Fraction with
Infinite Terms",
journal = j-FIB-QUART,
volume = "7",
number = "3",
pages = "275--276",
month = oct,
year = "1969",
CODEN = "FIBQAU",
ISSN = "0015-0517",
ISSN-L = "0015-0517",
bibdate = "Thu Oct 20 18:05:17 MDT 2011",
bibsource = "http://www.fq.math.ca/7-3.html;
https://www.math.utah.edu/pub/tex/bib/fibquart.bib;
https://www.math.utah.edu/pub/tex/bib/pi.bib",
URL = "http://www.fq.math.ca/Scanned/7-3/draim.pdf",
acknowledgement = ack-nhfb,
ajournal = "Fib. Quart",
fjournal = "The Fibonacci Quarterly",
journal-URL = "http://www.fq.math.ca/",
}
@Article{Stark:1969:CNA,
author = "E. L. Stark",
title = "Classroom Notes: Another Proof of the Formula $ \sum 1
/ k^2 = \pi^2 / 6 $",
journal = j-AMER-MATH-MONTHLY,
volume = "76",
number = "5",
pages = "552--553",
month = may,
year = "1969",
CODEN = "AMMYAE",
ISSN = "0002-9890 (print), 1930-0972 (electronic)",
ISSN-L = "0002-9890",
bibdate = "Mon Jun 28 12:39:24 MDT 1999",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib; JSTOR
database",
acknowledgement = ack-nhfb,
fjournal = "American Mathematical Monthly",
journal-URL = "https://www.jstor.org/journals/00029890.htm",
}
@Article{Moakes:1970:C,
author = "A. J. Moakes",
title = "The calculation of $ \pi $",
journal = j-MATH-GAZ,
volume = "54",
pages = "261--264",
year = "1970",
CODEN = "MAGAAS",
DOI = "https://doi.org/10.2307/3613778",
ISSN = "0025-5572",
bibdate = "Mon Apr 25 17:08:25 2011",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
ZMnumber = "0213.41904",
acknowledgement = ack-nhfb,
classmath = "*65D20 (Computation of special functions) 65A05
(Tables)",
fjournal = "Mathematical Gazette",
journal-URL = "http://www.m-a.org.uk/jsp/index.jsp?lnk=620",
}
@Article{Smeur:1970:VEA,
author = "A. J. E. M. Smeur",
title = "On the value equivalent to $ \pi $ in ancient
mathematical texts. {A} new interpretation",
journal = j-ARCH-HIST-EXACT-SCI,
volume = "6",
number = "4",
pages = "249--270",
month = jan,
year = "1970",
CODEN = "AHESAN",
DOI = "https://doi.org/10.1007/BF00417620",
ISSN = "0003-9519 (print), 1432-0657 (electronic)",
ISSN-L = "0003-9519",
MRclass = "Contributed Item",
MRnumber = "1554129",
bibdate = "Fri Feb 4 21:50:07 MST 2011",
bibsource = "http://springerlink.metapress.com/openurl.asp?genre=issue&issn=0003-9519&volume=6&issue=4;
https://www.math.utah.edu/pub/tex/bib/pi.bib",
URL = "http://www.springerlink.com/openurl.asp?genre=article&issn=0003-9519&volume=6&issue=4&spage=249",
acknowledgement = ack-nhfb,
fjournal = "Archive for History of Exact Sciences",
journal-URL = "http://link.springer.com/journal/407",
MRtitle = "On the value equivalent to {$\pi$} in ancient
mathematical texts. {A} new interpretation",
}
@Book{Beckmann:1971:H,
author = "Petr Beckmann",
title = "A History of $ \pi $",
publisher = pub-ST-MARTINS,
address = pub-ST-MARTINS:adr,
pages = "????",
year = "1971",
bibdate = "Sat Apr 23 09:43:28 2011",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
note = "Pi",
acknowledgement = ack-nhfb,
}
@Article{Choong:1971:RA,
author = "K. Y. Choong and D. E. Daykin and C. R. Rathbone",
title = "Rational Approximations to $ \pi $",
journal = j-MATH-COMPUT,
volume = "25",
number = "114",
pages = "387--392",
month = apr,
year = "1971",
CODEN = "MCMPAF",
ISSN = "0025-5718 (print), 1088-6842 (electronic)",
ISSN-L = "0025-5718",
bibdate = "Tue Oct 13 08:06:19 MDT 1998",
bibsource = "https://www.math.utah.edu/pub/tex/bib/mathcomp1970.bib;
https://www.math.utah.edu/pub/tex/bib/pi.bib; JSTOR
database",
note = "See errata \cite{Shanks:1976:TER}.",
URL = "http://www.ams.org/journals/mcom/1971-25-114/S0025-5718-1971-0300981-0",
acknowledgement = ack-nhfb,
fjournal = "Mathematics of Computation",
journal-URL = "http://www.ams.org/mcom/",
}
@Article{Lauro:1972:SDS,
author = "N. Lauro",
title = "Sulla distribuzione statistica delle cifre decimale di
$ \pi $. ({Italian}) [{On} the statistical distribution
of the decimal digits of $ \pi $]",
journal = "Studi Economici, Giannini, Napoli",
volume = "??",
number = "??",
pages = "77--93",
month = "????",
year = "1972",
bibdate = "Fri Jul 01 10:39:09 2011",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
acknowledgement = ack-nhfb,
language = "Italian",
remark = "Similar work to that of
\cite{Esmenjaud-Bonnardel:1965:ESD}.",
}
@Article{Papadimitriou:1973:CNS,
author = "Ioannis Papadimitriou",
title = "Classroom Notes: a Simple Proof of the Formula $
\sum^\infty_{k = 1} = \pi^2 / 6 $",
journal = j-AMER-MATH-MONTHLY,
volume = "80",
number = "4",
pages = "424--425",
month = apr,
year = "1973",
CODEN = "AMMYAE",
ISSN = "0002-9890 (print), 1930-0972 (electronic)",
ISSN-L = "0002-9890",
bibdate = "Mon Jun 28 12:37:07 MDT 1999",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib; JSTOR
database",
acknowledgement = ack-nhfb,
fjournal = "American Mathematical Monthly",
journal-URL = "https://www.jstor.org/journals/00029890.htm",
}
@Article{Fox:1975:FH,
author = "L. Fox and Linda Hayes",
title = "A further helping of $ \pi $",
journal = j-MATH-GAZ,
volume = "59",
number = "407",
pages = "38--40",
month = mar,
year = "1975",
CODEN = "MAGAAS",
DOI = "https://doi.org/10.2307/3616808",
ISSN = "0025-5572 (print), 2056-6328 (electronic)",
bibdate = "Tue Nov 14 08:15:28 2023",
bibsource = "https://www.math.utah.edu/pub/bibnet/authors/f/fox-leslie.bib;
https://www.math.utah.edu/pub/tex/bib/pi.bib",
URL = "https://www.jstor.org/stable/3616808",
acknowledgement = ack-nhfb,
author-dates = "Leslie Fox (30 September 1918--1 August 1992)",
fjournal = "Mathematical Gazette",
journal-URL = "http://www.jstor.org/journal/mathgaze",
remark = "Comment on Richardson's extrapolation methods for
computing $ \pi $ from successive inscribed polygons,
and report that ``The results are rather spectacular,
the two correct figures in $ p_{24} $ giving rise, with
a relatively trivial amount of extra arithmetic, to
nine correct figures in $ p'''_{24} $. Convergence
acceleration of this kind is obviously especially
valuable when computation of successive values of $ p_n
$, as here, is not trivial and is increasingly
contaminated with rounding errors.''",
}
@Article{Brent:1976:FMP,
author = "Richard P. Brent",
title = "Fast Multiple-Precision Evaluation of Elementary
Functions",
journal = j-J-ACM,
volume = "23",
number = "2",
pages = "242--251",
month = apr,
year = "1976",
CODEN = "JACOAH",
DOI = "https://doi.org/10.1145/321941.321944",
ISSN = "0004-5411 (print), 1557-735X (electronic)",
ISSN-L = "0004-5411",
MRclass = "68A20 (68A10)",
MRnumber = "52 \#16111",
MRreviewer = "Amnon Barak",
bibdate = "Wed Jan 15 18:12:53 MST 1997",
bibsource = "Compendex database;
garbo.uwasa.fi:/pc/doc-soft/fpbiblio.txt;
https://www.math.utah.edu/pub/tex/bib/pi.bib",
abstract = "Let $ f(x) $ be one of the usual elementary functions
($ \exp $, $ \log $, $ \arctan $, $ \sin $, $ \cosh $,
etc.), and let $ M(n) $ be the number of
single-precision operations required to multiply
$n$-bit integers. It is shown that $ f(x)$ can be
evaluated, with relative error $ O(2 - n)$, in $
O(M(n)l o g (n))$ operations as $ n \rightarrow \infty
$, for any floating-point number $x$ (with an $n$-bit
fraction) in a suitable finite interval. From the
Sch{\"o}nhage--Strassen bound on $ M(n)$, it follows
that an $n$-bit approximation to $ f(x)$ may be
evaluated in $ O(n \log_(n) \log \log (n))$ operations.
Special cases include the evaluation of constants such
as $ \pi $ $e$, and $ e^\pi $. The algorithms depend on
the theory of elliptic integrals, using the
arithmetic-geometric mean iteration and ascending
Landen transformations.",
acknowledgement = ack-nhfb,
classification = "723",
fjournal = "Journal of the Association for Computing Machinery",
journal-URL = "http://portal.acm.org/browse_dl.cfm?idx=J401",
journalabr = "J Assoc Comput Mach",
keywords = "computational complexity; computer arithmetic;
computer programming",
}
@InProceedings{Brent:1976:MPZ,
author = "Richard P. Brent",
title = "Multiple-precision zero-finding methods and the
complexity of elementary function evaluation",
crossref = "Traub:1976:ACC",
pages = "151--176",
year = "1976",
bibdate = "Tue Apr 26 09:42:05 2011",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
note = "Based on Interim Report ADA014059, Department of
Computer Science, Carnegie-Mellon University (July
1975), ii + 26 pages. See also \cite{Salamin:1976:CUA}
and update in \cite{Brent:2010:MPZ}.",
URL = "http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.119.3317;
http://wwwmaths.anu.edu.au/~brent/pub/pub028.html",
acknowledgement = ack-nhfb,
remark = "This paper contains a rediscovery of Salamin's formula
for finding $ \pi $ via the arithmetic-geometric
mean.",
}
@Article{Salamin:1976:CUA,
author = "Eugene Salamin",
title = "Computation of $ \pi $ Using Arithmetic-Geometric
Mean",
journal = j-MATH-COMPUT,
volume = "30",
number = "135",
pages = "565--570",
month = jul,
year = "1976",
CODEN = "MCMPAF",
ISSN = "0025-5718 (print), 1088-6842 (electronic)",
ISSN-L = "0025-5718",
MRclass = "10A30 (10A40 33A25)",
MRnumber = "0404124 (53 \#7928)",
MRreviewer = "I. John Zucker",
bibdate = "Tue Oct 13 08:06:19 MDT 1998",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib; JSTOR
database; MathSciNet database",
note = "See also \cite{Brent:1976:MPZ,Brent:2010:MPZ}.",
ZMnumber = "0345.10003",
acknowledgement = ack-nhfb,
classcodes = "B0290D (Functional analysis); C4120 (Functional
analysis)",
corpsource = "Charles Stark Draper Lab., Cambridge, MA, USA",
fjournal = "Mathematics of Computation",
journal-URL = "http://www.ams.org/mcom/",
keywords = "arithmetic geometric mean; convergence; elliptic
integrals; error analysis; fast Fourier transform
multiplication; function evaluation; Landen's;
Legendre's relation; numerical computation of pi;
transformation",
treatment = "A Application; T Theoretical or Mathematical",
}
@Article{Shanks:1976:TER,
author = "D. Shanks",
title = "Table errata: {``Regular continued fractions for $ \pi
$ and $ \gamma $'', (Math. Comp. {\bf 25} (1971), 403);
``Rational approximations to $ \pi $'' (ibid. {\bf 25}
(1971), 387--392) by K. Y. Choong, D. E. Daykin and C.
R. Rathbone}",
journal = j-MATH-COMPUT,
volume = "30",
number = "134",
pages = "381--381",
year = "1976",
CODEN = "MCMPAF",
DOI = "https://doi.org/10.1090/S0025-5718-1976-0386215-4",
ISSN = "0025-5718 (print), 1088-6842 (electronic)",
ISSN-L = "0025-5718",
MRclass = "65A05 (10-04 10F20)",
MRnumber = "0386215 (52 \#7073)",
bibdate = "Wed Jan 14 13:22:34 2015",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
URL = "http://www.ams.org/journals/mcom/1976-30-134/S0025-5718-1976-0386215-4",
acknowledgement = ack-nhfb,
fjournal = "Mathematics of Computation",
journal-URL = "http://www.ams.org/mcom/",
remark = "The second paper in the title is actually a review of
a report containing table of partial quotients for a
simple continued fraction for $ \pi $.",
}
@Book{Beckmann:1977:HP,
author = "Petr Beckmann",
title = "A History of $ \pi $",
publisher = pub-GOLEM,
address = pub-GOLEM:adr,
edition = "Fourth",
pages = "202",
year = "1977",
ISBN = "0-911762-18-3",
ISBN-13 = "978-0-911762-18-1",
LCCN = "QA484 .B4 1977",
bibdate = "Thu Sep 08 11:17:17 1994",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
note = "This book chronicles the story of the ultimate version
number of {\TeX}.",
acknowledgement = ack-nhfb,
}
@Article{Anderson:1978:F,
author = "Peter G. Anderson",
title = "On the Formula $ \pi = 2 \sum \arccot f_{2k + 1} $",
journal = j-FIB-QUART,
volume = "16",
number = "2",
pages = "118--??",
month = apr,
year = "1978",
CODEN = "FIBQAU",
ISSN = "0015-0517",
ISSN-L = "0015-0517",
bibdate = "Thu Oct 20 17:59:26 MDT 2011",
bibsource = "http://www.fq.math.ca/16-2.html;
https://www.math.utah.edu/pub/tex/bib/fibquart.bib;
https://www.math.utah.edu/pub/tex/bib/pi.bib",
URL = "http://www.fq.math.ca/Scanned/16-2/anderson.pdf",
acknowledgement = ack-nhfb,
ajournal = "Fib. Quart",
fjournal = "The Fibonacci Quarterly",
journal-URL = "http://www.fq.math.ca/",
}
@Article{Brent:1978:AMF,
author = "Richard P. Brent",
title = "{Algorithm 524}: {MP}, {A Fortran} Multiple-Precision
Arithmetic Package [{A1}]",
journal = j-TOMS,
volume = "4",
number = "1",
pages = "71--81",
month = mar,
year = "1978",
CODEN = "ACMSCU",
DOI = "https://doi.org/10.1145/355769.355776",
ISSN = "0098-3500 (print), 1557-7295 (electronic)",
ISSN-L = "0098-3500",
bibdate = "Tue Mar 09 10:35:50 1999",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
note = "See also
\cite{Brent:1979:RMF,Brent:1980:AIB,Smith:1998:AMP}.",
acknowledgement = ack-nhfb,
fjournal = "ACM Transactions on Mathematical Software",
journal-URL = "http://portal.acm.org/toc.cfm?idx=J782",
}
@Book{Solomon:1978:GP,
author = "Herbert Solomon",
title = "Geometric probability",
volume = "28",
publisher = pub-SIAM,
address = pub-SIAM:adr,
pages = "vi + 174",
year = "1978",
ISBN = "0-89871-025-1 (paperback)",
ISBN-13 = "978-0-89871-025-0 (paperback)",
LCCN = "QA273.5 .S64 1978; QA273.5 .S65",
bibdate = "Tue Apr 29 20:39:05 MDT 2014",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib;
https://www.math.utah.edu/pub/tex/bib/probstat1970.bib;
prodorbis.library.yale.edu:7090/voyager;
z3950.loc.gov:7090/Voyager",
series = "CBMS-NSF regional conference series in applied
mathematics",
URL = "http://epubs.siam.org/ebooks/siam/cbms-nsf_regional_conference_series_in_applied_mathematics/cb28",
abstract = "Topics include: ways modern statistical procedures can
yield estimates of pi more precisely than the original
Buffon procedure traditionally used; the question of
density and measure for random geometric elements that
leave probability and expectation statements invariant
under translation and rotation; the number of random
line intersections in a plane and their angles of
intersection; developments due to W.L. Stevens's
ingenious solution for evaluating the probability that
n random arcs of size a cover a unit circumference
completely; the development of M.W. Crofton's mean
value theorem and its applications in classical
problems; and an interesting problem in geometrical
probability presented by a karyograph.",
acknowledgement = ack-nhfb,
subject = "Geometric probabilities",
tableofcontents = "Buffon needle problem, extensions, and estimation
of pi \\
Density and measure for random geometric elements \\
Random lines in the plane and applications \\
Covering a circle circumference and a sphere surface
\\
Crofton's theorem and Sylvester's problem in two and
three dimensions \\
Random chords in the circle and the sphere",
xxpages = "vii + 172",
}
@Article{Brent:1979:RMF,
author = "R. P. Brent",
title = "Remark on ``{Algorithm} 524: {MP}, {A Fortran}
Multiple-Precision Arithmetic Package [{A1}]''",
journal = j-TOMS,
volume = "5",
number = "4",
pages = "518--519",
month = dec,
year = "1979",
CODEN = "ACMSCU",
DOI = "https://doi.org/10.1145/355853.355868",
ISSN = "0098-3500 (print), 1557-7295 (electronic)",
ISSN-L = "0098-3500",
bibdate = "Tue Mar 09 10:35:42 1999",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
note = "See
\cite{Brent:1978:AMF,Brent:1980:AIB,Smith:1998:AMP}.",
acknowledgement = ack-nhfb,
fjournal = "ACM Transactions on Mathematical Software",
journal-URL = "http://portal.acm.org/toc.cfm?idx=J782",
}
@Article{Ferguson:1979:GEA,
author = "H. R. P. Ferguson and R. W. Forcade",
title = "Generalization of the {Euclidean} Algorithm for Real
Numbers to All Dimensions Higher than Two",
journal = j-BULL-AMS-N-S,
volume = "1",
number = "??",
pages = "912--914",
month = "????",
year = "1979",
CODEN = "BAMOAD",
DOI = "https://doi.org/10.1090/S0273-0979-1979-14691-3",
ISSN = "0273-0979 (print), 1088-9485 (electronic)",
ISSN-L = "0273-0979",
MRclass = "10E45, 10F10, 10F20 (primary); 10F37, 12A10, 10H05,
02E10 (secondary)",
MRnumber = "546316, MR 80i:11039",
bibdate = "Tue Apr 26 16:14:10 2011",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
abstract = "A construction using integral matrices with
determinant $ \pm 1 $ is given which has as corollaries
generalizations of classical theorems of Dirichlet and
Kronecker. This construction yields a geometrically
convergent algorithm successfully generalizing the
Euclidean algorithm to finite sets of real numbers.
Applied to such a set this algorithm terminates if and
only if the set is integrally linearly dependent and
the algorithm gives absolute simultaneous integral
approximations if and only if the set is integrally
linearly independent. This development applies to
complex numbers, can be used to give proofs of
irreducibility of polynomials and yields effective
lower bounds on heights of integral relations.",
acknowledgement = ack-nhfb,
fjournal = "Bulletin of the American Mathematical Society",
}
@Article{Miel:1979:CNA,
author = "George Miel",
title = "Classroom Notes: An Algorithm for the Calculation of $
\pi $",
journal = j-AMER-MATH-MONTHLY,
volume = "86",
number = "8",
pages = "694--697",
month = oct,
year = "1979",
CODEN = "AMMYAE",
DOI = "https://doi.org/10.2307/2321304",
ISSN = "0002-9890 (print), 1930-0972 (electronic)",
ISSN-L = "0002-9890",
MRclass = "65D20",
MRnumber = "80k:65021",
MRreviewer = "Gerhard Merz",
bibdate = "Mon Jun 28 12:39:33 MDT 1999",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib; JSTOR
database",
acknowledgement = ack-nhfb,
fjournal = "American Mathematical Monthly",
journal-URL = "https://www.jstor.org/journals/00029890.htm",
}
@Unpublished{Bergman:1980:NFF,
author = "G. Bergman",
title = "Notes on {Ferguson} and {Forcade}'s generalized
{Euclidean} algorithm",
month = nov,
year = "1980",
bibdate = "Tue Apr 26 17:07:21 2011",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
note = "Unpublished notes, University of California at
Berkeley.",
acknowledgement = ack-nhfb,
remark = "See \cite{Ferguson:1979:GEA}.",
}
@Article{Brent:1980:AIB,
author = "Richard P. Brent and Judith A. Hooper and J. Michael
Yohe",
title = "An {AUGMENT} Interface for {Brent}'s Multiple
Precision Arithmetic Package",
journal = j-TOMS,
volume = "6",
number = "2",
pages = "146--149",
month = jun,
year = "1980",
CODEN = "ACMSCU",
DOI = "https://doi.org/10.1145/355887.355889",
ISSN = "0098-3500 (print), 1557-7295 (electronic)",
ISSN-L = "0098-3500",
bibdate = "Tue Mar 09 10:35:33 1999",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
note = "See
\cite{Brent:1978:AMF,Brent:1979:RMF,Smith:1998:AMP}.",
acknowledgement = ack-nhfb,
fjournal = "ACM Transactions on Mathematical Software",
journal-URL = "http://portal.acm.org/toc.cfm?idx=J782",
keywords = "arithmetic; AUGMENT interface; extended precision;
floating point; multiple precision; portable software;
precompiler interface; software package",
}
@Article{Baxter:1981:UPE,
author = "L. Baxter",
title = "Unsolved Problems: Are $ \pi, e $, and $ \surd 2 $
Equally Difficult to Compute?",
journal = j-AMER-MATH-MONTHLY,
volume = "88",
number = "1",
pages = "50--51",
month = jan,
year = "1981",
CODEN = "AMMYAE",
ISSN = "0002-9890 (print), 1930-0972 (electronic)",
ISSN-L = "0002-9890",
bibdate = "Mon Jun 28 12:36:14 MDT 1999",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib; JSTOR
database",
acknowledgement = ack-nhfb,
fjournal = "American Mathematical Monthly",
journal-URL = "https://www.jstor.org/journals/00029890.htm",
}
@Article{Cohen:1981:JWM,
author = "G. L. Cohen and A. G. Shannon",
title = "{John Ward}'s method for the calculation of pi [$ \pi
$ ]",
journal = j-HIST-MATH,
volume = "8",
number = "2",
pages = "133--144",
month = may,
year = "1981",
CODEN = "HIMADS",
DOI = "https://doi.org/10.1016/0315-0860(81)90025-2",
ISSN = "0315-0860 (print), 1090-249X (electronic)",
ISSN-L = "0315-0860",
MRclass = "01A50",
MRnumber = "618366 (83d:01021)",
MRreviewer = "Garry J. Tee",
bibdate = "Wed Jun 26 06:17:24 MDT 2013",
bibsource = "https://www.math.utah.edu/pub/tex/bib/histmath.bib;
https://www.math.utah.edu/pub/tex/bib/pi.bib; MathSciNet
database",
URL = "http://www.sciencedirect.com/science/article/pii/0315086081900252",
abstract = "What may be the last attempt to use geometric methods
to calculate pi is found in a textbook published in
England in 1707. The underlying algebraic and numerical
methods are analyzed in this paper.",
acknowledgement = ack-nhfb,
fjournal = "Historia Mathematica",
journal-URL = "http://www.sciencedirect.com/science/journal/03150860",
}
@Unpublished{Forcade:1981:BA,
author = "Rodney W. Forcade",
title = "{Brun}'s algorithm",
pages = "1--27",
month = nov,
year = "1981",
bibdate = "Tue Apr 26 17:14:28 2011",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
note = "Unpublished manuscript",
acknowledgement = ack-nhfb,
}
@Article{Ferguson:1982:MEA,
author = "H. R. P. Ferguson and R. W. Forcade",
title = "Multidimensional {Euclidean} Algorithms",
journal = j-J-REINE-ANGEW-MATH,
volume = "334",
number = "??",
pages = "171--181",
month = "????",
year = "1982",
CODEN = "JRMAA8",
ISSN = "0075-4102",
ISSN-L = "0075-4102",
MRnumber = "MR 84d:10015",
bibdate = "Tue Apr 26 16:22:54 2011",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
URL = "http://www.ams.org/mathscinet-getitem?mr=84d:10015",
abstract = "The authors construct an iterative algorithm for
$n$-tuples (the $ \mathrm {GL}_n(Z)$ algorithm $
A_n(b)$), generalizing both the terminating and the
approximating features of the Euclidean algorithm. The
algorithm depends on a parameter $b$ in the interval $
(1 / 2, 1)$, an $n$-tuple $ x \in \mathbf {R}^n$ and a
hyperplane. This algorithm generates a sequence of
matrices $ M_k$ such that one of the following holds:
(1) Termination: There exists a $k$ such that a column
of $ M_k$ is an integral relation among the entries of
$x$, or (2) Approximation: For every $ \epsilon > 0$
there exists an integer $ K \geq 1$ such that for each
$ k \geq K$ the rows of $ M_k^{-1}$ give $n$ linearly
independent lattice points in $ Z^n$ each within a
distance of the line determined by $x$. Some
applications of this algorithm are given in the end of
the paper.",
acknowledgement = ack-nhfb,
fjournal = "Journal f{\"u}r die reine und angewandte Mathematik",
keywords = "precursor of PSLQ algorithm",
}
@InCollection{Newman:1982:RAV,
author = "Donald J. Newman",
booktitle = "Lectures on approximation and value distribution",
title = "Rational approximation versus fast computer methods",
volume = "79",
publisher = "Presses de l'universit{\'e} de Montr{\'e}al",
address = "Montr{\'e}al, QC H3C 3J7, Canada",
pages = "149--174",
year = "1982",
MRclass = "41A20 41A25 65D20 41A21",
MRnumber = "MR0654686 (83e:41021)",
bibdate = "Thu Jun 09 15:53:59 2016",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
series = "S{\'e}minaire de Math{\'e}matiques Sup{\'e}rieures",
ZMnumber = "0506.41014",
acknowledgement = ack-nhfb,
}
@Article{Borwein:1983:VRC,
author = "Jonathan M. Borwein and Peter B. Borwein",
title = "A very rapidly convergent product expansion for $ \pi
$ [pi]",
journal = j-BIT,
volume = "23",
number = "4",
pages = "538--540",
month = dec,
year = "1983",
CODEN = "BITTEL, NBITAB",
DOI = "https://doi.org/10.1007/BF01933626",
ISSN = "0006-3835 (print), 1572-9125 (electronic)",
ISSN-L = "0006-3835",
MRclass = "65B99",
MRnumber = "85h:65011",
bibdate = "Wed Jan 4 18:52:18 MST 2006",
bibsource = "http://springerlink.metapress.com/openurl.asp?genre=issue&issn=0006-3835&volume=23&issue=4;
https://www.math.utah.edu/pub/tex/bib/bit.bib;
https://www.math.utah.edu/pub/tex/bib/pi.bib",
URL = "http://www.springerlink.com/openurl.asp?genre=article&issn=0006-3835&volume=23&issue=4&spage=538",
acknowledgement = ack-nhfb,
author-dates = "Jonathan Michael Borwein (20 May 1951--2 August
2016)",
fjournal = "BIT",
journal-URL = "http://link.springer.com/journal/10543",
ORCID-numbers = "Borwein, Jonathan/0000-0002-1263-0646",
}
@TechReport{Kanada:1983:CDP,
author = "Y. Kanada and Y. Tamura and S. Yoshino and Y. Ushiro",
title = "Calculation of $ \pi $ to 10,013,395 Decimal Places
Based on the {Gauss--Legendre} Algorithm and {Gauss}
Arctangent Relation",
type = "Technical report",
number = "CCUT-TR-84-01",
institution = "Computer Centre, University of Tokyo",
address = "Bunkyo-ky, Yayoi 2-11-16, Tokyo 113, Japan",
month = dec,
year = "1983",
bibdate = "Mon Jul 18 17:50:42 2005",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
acknowledgement = ack-nhfb,
}
@TechReport{Tamura:1983:CDB,
author = "Y. Tamura and Y. Kanada",
title = "Calculation of $ \pi $ to 4,194,293 Decimals Based on
the {Gauss--Legendre} Algorithm",
type = "Technical report",
number = "CCUT-TR-83-01",
institution = "Computer Centre, University of Tokyo",
address = "Tokyo, Japan",
month = may,
year = "1983",
bibdate = "Mon Jul 18 17:46:12 2005",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
acknowledgement = ack-nhfb,
}
@Article{Borwein:1984:CHO,
author = "J. M. Borwein and P. B. Borwein",
title = "Cubic and higher order algorithms for $ \pi $",
journal = j-CAN-MATH-BULL,
volume = "27",
number = "??",
pages = "436--443",
month = "????",
year = "1984",
CODEN = "CMBUA3",
DOI = "https://doi.org/10.4153/CMB-1984-067-7",
ISSN = "0008-4395 (print), 1496-4287 (electronic)",
ISSN-L = "0008-4395",
bibdate = "Thu Sep 8 10:05:21 MDT 2011",
bibsource = "http://cms.math.ca/cmb/v27/;
https://www.math.utah.edu/pub/tex/bib/pi.bib",
acknowledgement = ack-nhfb,
author-dates = "Jonathan Michael Borwein (20 May 1951--2 August
2016)",
fjournal = "Canadian mathematical bulletin = Bulletin canadien de
math{\'e}matiques",
journal-URL = "http://cms.math.ca/cmb/",
ORCID-numbers = "Borwein, Jonathan/0000-0002-1263-0646",
}
@InProceedings{Borwein:1984:EAO,
author = "J. M. Borwein and P. B. Borwein",
title = "Explicit algebraic $n$ th order approximations to pi",
crossref = "Singh:1984:ATS",
volume = "136",
pages = "247--256",
year = "1984",
DOI = "https://doi.org/10.1007/978-94-009-6466-2_12",
MRclass = "65D20",
MRnumber = "786845",
bibdate = "Thu Aug 11 09:36:22 MDT 2016",
bibsource = "https://www.math.utah.edu/pub/bibnet/authors/b/borwein-jonathan-m.bib;
https://www.math.utah.edu/pub/tex/bib/pi.bib",
URL = "http://link.springer.com/chapter/10.1007/978-94-009-6466-2_12",
acknowledgement = ack-nhfb,
author-dates = "Jonathan Michael Borwein (20 May 1951--2 August
2016)",
ORCID-numbers = "Borwein, Jonathan/0000-0002-1263-0646",
}
@Article{Newman:1984:SAS,
author = "Morris Newman and Daniel Shanks",
title = "On a sequence arising in series for $ \pi $",
journal = j-MATH-COMPUT,
volume = "42",
number = "165",
pages = "199--217",
month = jan,
year = "1984",
CODEN = "MCMPAF",
ISSN = "0025-5718 (print), 1088-6842 (electronic)",
ISSN-L = "0025-5718",
MRclass = "11Y35 (11F11)",
MRnumber = "85k:11069",
MRreviewer = "D. H. Lehmer",
bibdate = "Tue Oct 13 08:06:19 MDT 1998",
bibsource = "https://www.math.utah.edu/pub/tex/bib/mathcomp1980.bib;
https://www.math.utah.edu/pub/tex/bib/pi.bib; JSTOR
database",
acknowledgement = ack-nhfb,
classcodes = "B0210 (Algebra); C1110 (Algebra)",
corpsource = "Dept. of Maths., Univ. of California, Santa Barbara,
CA, USA",
fjournal = "Mathematics of Computation",
journal-URL = "http://www.ams.org/mcom/",
keywords = "adic numbers; cubic recurrences; p-; positive
integers; rational sequence; sequences; series; series
(mathematics)",
treatment = "T Theoretical or Mathematical",
}
@InProceedings{Haastad:1985:PTA,
author = "J. H{\aa}stad and B. Helfrich and J. Lagarias and C.
P. Schnorr",
title = "Polynomial time algorithms for finding integer
relations among real numbers",
crossref = "Monien:1986:SAS",
publisher = pub-SV,
address = pub-SV:adr,
pages = "105--118",
year = "1985",
DOI = "https://doi.org/10.1007/3-540-16078-7_69",
bibdate = "Tue Apr 26 16:03:29 2011",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
abstract = "We present algorithms, which when given a real vector
$ x 2^{\frac {{n - 2}}{2}} $ times longer than the
length of the shortest relation for $x$. Given a
rational input $ x \in Q^n$, this algorithm halts in
polynomially many bit operations. The basic algorithm
of this kind is due to Ferguson and Forcade (1979) and
is closely related to the Lov{\`a}sz (1982) lattice
basis reduction algorithm.",
acknowledgement = ack-nhfb,
}
@Article{Montgomery:1985:MMT,
author = "Peter L. Montgomery",
title = "Modular Multiplication Without Trial Division",
journal = j-MATH-COMPUT,
volume = "44",
number = "170",
pages = "519--521",
month = apr,
year = "1985",
CODEN = "MCMPAF",
ISSN = "0025-5718 (print), 1088-6842 (electronic)",
ISSN-L = "0025-5718",
MRclass = "11Y16",
MRnumber = "86e:11121",
bibdate = "Tue Oct 13 08:06:19 MDT 1998",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib; JSTOR
database",
URL = "http://www.jstor.org/stable/2007970",
abstract = "Let $ N > 1 $. We present a method for multiplying two
integers (called $N$-residues) modulo $N$ while
avoiding division by $ N. N$-residues are represented
in a nonstandard way, so this method is useful only if
several computations are done modulo one $N$. The
addition and subtraction algorithms are unchanged.",
acknowledgement = ack-nhfb,
classcodes = "C1160 (Combinatorial mathematics); C5230 (Digital
arithmetic methods); C6130 (Data handling techniques)",
corpsource = "Syst. Dev. Corp., Santa Monica, CA, USA",
fjournal = "Mathematics of Computation",
journal-URL = "http://www.ams.org/mcom/",
keywords = "digital arithmetic; integer; integer arithmetic;
modular arithmetic; modular multiplication;
multiplication; N-residue; N-residue arithmetic; number
theory",
treatment = "T Theoretical or Mathematical",
}
@Article{Newman:1985:SVF,
author = "D. J. Newman",
title = "A simplified version of the fast algorithms of {Brent}
and {Salamin}",
journal = j-MATH-COMPUT,
volume = "44",
number = "169",
pages = "207--210",
month = jan,
year = "1985",
CODEN = "MCMPAF",
ISSN = "0025-5718 (print), 1088-6842 (electronic)",
ISSN-L = "0025-5718",
MRclass = "65D20",
MRnumber = "86e:65030",
MRreviewer = "Walter Gautschi",
bibdate = "Tue Oct 13 08:06:19 MDT 1998",
bibsource = "https://www.math.utah.edu/pub/tex/bib/mathcomp1980.bib;
https://www.math.utah.edu/pub/tex/bib/pi.bib; JSTOR
database",
acknowledgement = ack-nhfb,
classcodes = "B0290F (Interpolation and function approximation);
C4130 (Interpolation and function approximation)",
journal-URL = "http://www.ams.org/mcom/",
keywords = "exponential; fast algorithms; function approximation;
function approximations; Gauss arithmetic-geometric
process; pi",
treatment = "T Theoretical or Mathematical",
}
@Article{Borwein:1986:ECI,
author = "J. M. Borwein and P. B. Borwein",
title = "An explicit cubic iteration for $ \pi $",
journal = j-BIT,
volume = "26",
number = "1",
pages = "123--126",
month = mar,
year = "1986",
CODEN = "BITTEL, NBITAB",
DOI = "https://doi.org/10.1007/BF01939368",
ISSN = "0006-3835 (print), 1572-9125 (electronic)",
ISSN-L = "0006-3835",
MRclass = "11Y60 (65D20)",
MRnumber = "87e:11144",
MRreviewer = "Duncan A. Buell",
bibdate = "Wed Jan 4 18:52:19 MST 2006",
bibsource = "http://springerlink.metapress.com/openurl.asp?genre=issue&issn=0006-3835&volume=26&issue=1;
https://www.math.utah.edu/pub/tex/bib/bit.bib;
https://www.math.utah.edu/pub/tex/bib/pi.bib",
URL = "http://www.springerlink.com/openurl.asp?genre=article&issn=0006-3835&volume=26&issue=1&spage=123",
acknowledgement = ack-nhfb,
author-dates = "Jonathan Michael Borwein (20 May 1951--2 August
2016)",
fjournal = "BIT",
journal-URL = "http://link.springer.com/journal/10543",
ORCID-numbers = "Borwein, Jonathan/0000-0002-1263-0646",
}
@Article{Borwein:1986:MQC,
author = "J. M. Borwein and P. B. Borwein",
title = "More Quadratically Converging Algorithms for $ \pi $",
journal = j-MATH-COMPUT,
volume = "46",
number = "173",
pages = "247--253",
month = jan,
year = "1986",
CODEN = "MCMPAF",
ISSN = "0025-5718 (print), 1088-6842 (electronic)",
ISSN-L = "0025-5718",
MRclass = "65D20",
MRnumber = "87e:65014",
MRreviewer = "M. M. Chawla",
bibdate = "Tue Oct 13 08:06:19 MDT 1998",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib; JSTOR
database",
acknowledgement = ack-nhfb,
author-dates = "Jonathan Michael Borwein (20 May 1951--2 August
2016)",
classcodes = "B0290F (Interpolation and function approximation);
B0290Z (Other numerical methods); C4130 (Interpolation
and function approximation); C4190 (Other numerical
methods)",
corpsource = "Dalhousie Univ., Halifax, NS, Canada",
fjournal = "Mathematics of Computation",
journal-URL = "http://www.ams.org/mcom/",
keywords = "arithmetic-geometric mean iteration; complete
elliptic; convergence of numerical methods;
Gauss--Legendre iteration; geometry; integrals;
iterative; Legendre formula; methods; pi evaluation;
quadratically converging algorithms",
ORCID-numbers = "Borwein, Jonathan/0000-0002-1263-0646",
treatment = "T Theoretical or Mathematical",
}
@Article{Ferguson:1986:SPE,
author = "H. R. P. Ferguson",
title = "A Short Proof of the Existence of Vector {Euclidean}
Algorithms",
journal = j-PROC-AM-MATH-SOC,
volume = "97",
number = "??",
pages = "8--10",
month = "??",
year = "1986",
CODEN = "PAMYAR",
ISSN = "0002-9939 (print), 1088-6826 (electronic)",
ISSN-L = "0002-9939",
MRnumber = "MR 87k:11080",
bibdate = "Tue Apr 26 16:19:39 2011",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
URL = "http://www.ams.org/mathscinet-getitem?mr=87k:11080",
acknowledgement = ack-nhfb,
fjournal = "Proceedings of the American Mathematical Society",
}
@Article{Hancl:1986:NSP,
author = "Jaroslav Han{\v{c}}l",
title = "Notes: {A} Simple Proof of the Irrationality of $
\pi^4 $",
journal = j-AMER-MATH-MONTHLY,
volume = "93",
number = "5",
pages = "374--375",
month = may,
year = "1986",
CODEN = "AMMYAE",
ISSN = "0002-9890 (print), 1930-0972 (electronic)",
ISSN-L = "0002-9890",
MRclass = "11J72",
MRnumber = "87g:11084",
MRreviewer = "Vichian Laohakosol",
bibdate = "Mon Jun 28 12:38:20 MDT 1999",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib; JSTOR
database",
acknowledgement = ack-nhfb,
fjournal = "American Mathematical Monthly",
journal-URL = "https://www.jstor.org/journals/00029890.htm",
}
@Article{Matiyasevich:1986:NNF,
author = "Yuri V. Matiyasevich",
title = "Notes: {A} New Formula for $ \pi $",
journal = j-AMER-MATH-MONTHLY,
volume = "93",
number = "8",
pages = "631--635",
month = oct,
year = "1986",
CODEN = "AMMYAE",
ISSN = "0002-9890 (print), 1930-0972 (electronic)",
ISSN-L = "0002-9890",
bibdate = "Mon Jun 28 12:38:26 MDT 1999",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib; JSTOR
database",
acknowledgement = ack-nhfb,
fjournal = "American Mathematical Monthly",
journal-URL = "https://www.jstor.org/journals/00029890.htm",
}
@Article{Parks:1986:NOI,
author = "Alan E. Parks",
title = "Notes: $ \pi, e $, and Other Irrational Numbers",
journal = j-AMER-MATH-MONTHLY,
volume = "93",
number = "9",
pages = "722--723",
month = nov,
year = "1986",
CODEN = "AMMYAE",
ISSN = "0002-9890 (print), 1930-0972 (electronic)",
ISSN-L = "0002-9890",
MRclass = "11J72",
MRnumber = "87j:11068",
bibdate = "Mon Jun 28 12:38:29 MDT 1999",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib; JSTOR
database",
acknowledgement = ack-nhfb,
fjournal = "American Mathematical Monthly",
journal-URL = "https://www.jstor.org/journals/00029890.htm",
}
@Unpublished{Bernstein:1987:NFA,
author = "Daniel J. Bernstein",
title = "New fast algorithms for $ \pi $ and $e$",
pages = "21",
year = "1987",
bibdate = "Mon Dec 31 16:56:43 2012",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
note = "Fifth-place paper for the nationwide 1987 Westinghouse
Science Talent Search. Distributed at the Ramanujan
Centenary Conference. The Web site has only JPEG images
of a document scan.",
URL = "http://cr.yp.to/bib/1987/bernstein.html",
acknowledgement = ack-nhfb,
}
@Article{Choe:1987:NEP,
author = "Boo Rim Choe",
title = "Notes: An Elementary Proof of $ \sum^\infty_{n = 1} 1
/ n^2 = \pi^2 / 6 $",
journal = j-AMER-MATH-MONTHLY,
volume = "94",
number = "7",
pages = "662--663",
month = aug # "\slash " # sep,
year = "1987",
CODEN = "AMMYAE",
ISSN = "0002-9890 (print), 1930-0972 (electronic)",
ISSN-L = "0002-9890",
MRclass = "40A25",
MRnumber = "935 853",
bibdate = "Mon Jun 28 12:38:46 MDT 1999",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib; JSTOR
database",
acknowledgement = ack-nhfb,
fjournal = "American Mathematical Monthly",
journal-URL = "https://www.jstor.org/journals/00029890.htm",
}
@Article{Edgar:1987:PDE,
author = "G. A. Edgar",
title = "Pi: Difficult or easy? {Mathematical} considerations
for the multidigit computation of pi",
journal = j-MATH-MAG,
volume = "60",
pages = "141--150",
year = "1987",
CODEN = "MAMGA8",
ISSN = "0025-570X",
ISSN-L = "0025-570X",
bibdate = "Mon Apr 25 18:01:33 2011",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
ZMnumber = "0627.65016",
abstract = "The author discusses an introduction to the
computational complexity concerning the multi-digit
computation of the numbers $ \pi $, $e$ and other few
mathematical constants. He considers only power series,
and no treatment on the acceleration of convergence, or
other rapidly converging procedures to compute the
above constants.",
acknowledgement = ack-nhfb,
classmath = "*65D20 (Computation of special functions) 65B10
(Summation of series) 68Q25 (Analysis of algorithms and
problem complexity)",
fjournal = "Mathematics Magazine",
journal-URL = "http://www.maa.org/pubs/mathmag.html",
keywords = "computational complexity; multi-digit computation; no
convergence acceleration; number e; number pi; power
series summation",
language = "English",
reviewer = "S. Hitotumatu",
}
@Article{Ferguson:1987:NIA,
author = "H. R. P. Ferguson",
title = "A Non-Inductive {$ \mathrm {GL}(n, Z) $} Algorithm
that Constructs Linear Relations for $n$ {$Z$}-Linearly
Dependent Real Numbers",
journal = j-J-ALG,
volume = "8",
number = "??",
pages = "131--145",
month = "????",
year = "1987",
CODEN = "JOALDV",
ISSN = "0196-6774 (print), 1090-2678 (electronic)",
ISSN-L = "0196-6774",
MRnumber = "MR 88h:11096",
bibdate = "Tue Apr 26 16:16:39 2011",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
URL = "http://www.ams.org/mathscinet-getitem?mr=88h:11096",
acknowledgement = ack-nhfb,
fjournal = "Journal of Algorithms",
journal-URL = "http://www.sciencedirect.com/science/journal/01966774",
keywords = "precursor of PSLQ algorithm",
}
@Article{Almkvist:1988:GLR,
author = "Gert Almkvist and Bruce Berndt",
title = "{Gauss}, {Landen}, {Ramanujan}, the
Arithmetic-Geometric Mean, Ellipses, $ \pi $, and the
{Ladies Diary}",
journal = j-AMER-MATH-MONTHLY,
volume = "95",
number = "7",
pages = "585--608",
month = aug # "\slash " # sep,
year = "1988",
CODEN = "AMMYAE",
ISSN = "0002-9890 (print), 1930-0972 (electronic)",
ISSN-L = "0002-9890",
MRclass = "01A50 (01A55 01A60 33A25)",
MRnumber = "89j:01028",
MRreviewer = "R. A. Askey",
bibdate = "Mon Jun 28 12:39:09 MDT 1999",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib; JSTOR
database",
acknowledgement = ack-nhfb,
fjournal = "American Mathematical Monthly",
journal-URL = "https://www.jstor.org/journals/00029890.htm",
}
@Article{Bailey:1988:CDD,
author = "David H. Bailey",
title = "The Computation of $ \pi $ to 29,360,000 Decimal
Digits Using {Borweins}' Quartically Convergent
Algorithm",
journal = j-MATH-COMPUT,
volume = "50",
number = "181",
pages = "283--296",
month = jan,
year = "1988",
CODEN = "MCMPAF",
DOI = "https://doi.org/10.1090/S0025-5718-1988-0917836-3;
https://doi.org/10.2307/2007932",
ISSN = "0025-5718 (print), 1088-6842 (electronic)",
ISSN-L = "0025-5718",
MRclass = "11Y60 (11-04 11K16 65-04)",
MRnumber = "88m:11114",
MRreviewer = "A. J. van der Poorten",
bibdate = "Tue Oct 13 08:06:19 MDT 1998",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib;
https://www.davidhbailey.com/dhbpapers/pi.pdf; JSTOR
database",
acknowledgement = ack-nhfb,
classcodes = "C1140Z (Other and miscellaneous); C1160 (Combinatorial
mathematics); C4130 (Interpolation and function
approximation); C5470 (Performance evaluation and
testing); C7310 (Mathematics)",
corpsource = "NASA Ames Res. Centre, Moffet Field, CA, USA",
fjournal = "Mathematics of Computation",
journal-URL = "http://www.ams.org/mcom/",
keywords = "Borwein quartically convergent algorithm; computation
of pi; computer testing; Cray 2 computer test; decimal
expansion; elliptic integrals; iterative methods;
mathematics computing; multiprecision arithmetic;
number theory; prime modulus; series (mathematics);
statistical analyses; statistical analysis; transform",
ORCID-numbers = "Bailey, David H./0000-0002-7574-8342",
treatment = "X Experimental",
}
@Article{Bailey:1988:NRT,
author = "David H. Bailey",
title = "Numerical Results on the Transcendence of Constants
Involving $ \pi $, $e$, and {Euler}'s Constant",
journal = j-MATH-COMPUT,
volume = "50",
number = "181",
pages = "275--281",
month = jan,
year = "1988",
CODEN = "MCMPAF",
ISSN = "0025-5718 (print), 1088-6842 (electronic)",
ISSN-L = "0025-5718",
MRclass = "11J81 (11Y60)",
MRnumber = "88m:11056",
MRreviewer = "David Lee Hilliker",
bibdate = "Tue Oct 13 08:06:19 MDT 1998",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib; JSTOR
database",
URL = "http://www.ams.org/mathscinet-getitem?mr=88m:11056",
acknowledgement = ack-nhfb,
classcodes = "C1160 (Combinatorial mathematics); C7310
(Mathematics)",
corpsource = "NASA Ames Res. Centre, Moffet Field, CA, USA",
fjournal = "Mathematics of Computation",
journal-URL = "http://www.ams.org/mcom/",
keywords = "Cray-2 supercomputer; e; Euler constant; exponential
constant; Forcade algorithm; mathematics computing;
multiprecision arithmetic; number theory; pi; recursive
Ferguson-; transcendental constants",
ORCID-numbers = "Bailey, David H./0000-0002-7574-8342",
treatment = "T Theoretical or Mathematical; X Experimental",
}
@Article{Ferguson:1988:PNI,
author = "Helaman Ferguson",
title = "{PSOS}: a new integral relation finding algorithm
involving partial sums of squares and no square roots",
journal = "Abstracts of papers presented to the {American
Mathematical Society}",
volume = "9",
number = "56 (88T-11-75)",
pages = "214--214",
month = mar,
year = "1988",
bibdate = "Tue Apr 26 17:13:15 2011",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
acknowledgement = ack-nhfb,
}
@Article{Hurley:1988:RCP,
author = "Donal Hurley",
title = "Recent computations of $ \pi $",
journal = "Irish Math. Soc. Bull.",
volume = "21",
number = "??",
pages = "38--44",
year = "1988",
ISSN = "0791-5578",
MRclass = "11Y60 (01A50 01A55 01A60 11-03)",
MRnumber = "988289 (90e:11194)",
MRreviewer = "Kenneth A. Jukes",
bibdate = "Mon Apr 25 16:20:53 2011",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
acknowledgement = ack-nhfb,
fjournal = "Irish Mathematical Society Bulletin",
keywords = "agm (arithmetic-geometric mean); Brent--Salamin
algorithm (1976); Johann Dase (1824--1861); John Machin
(1680--1752)",
remark = "No issues before 1995 are available online at
http://www.maths.tcd.ie/pub/ims/bulletin/index.php.",
}
@Article{Jami:1988:HCD,
author = "Catherine Jami",
title = "Une histoire chinoise du ``nombre $ \pi $''.
({French}) [{A} {Chinese} history of the ``number $ \pi
$'']",
journal = j-ARCH-HIST-EXACT-SCI,
volume = "38",
number = "1",
pages = "39--50",
month = mar,
year = "1988",
CODEN = "AHESAN",
DOI = "https://doi.org/10.1007/BF00329979",
ISSN = "0003-9519 (print), 1432-0657 (electronic)",
ISSN-L = "0003-9519",
MRclass = "01A25",
MRnumber = "925728 (90j:01012)",
MRreviewer = "J. Friberg",
bibdate = "Fri Feb 4 21:50:25 MST 2011",
bibsource = "http://springerlink.metapress.com/openurl.asp?genre=issue&issn=0003-9519&volume=38&issue=1;
https://www.math.utah.edu/pub/tex/bib/pi.bib",
URL = "http://www.springerlink.com/openurl.asp?genre=article&issn=0003-9519&volume=38&issue=1&spage=39",
acknowledgement = ack-nhfb,
fjournal = "Archive for History of Exact Sciences",
journal-URL = "http://link.springer.com/journal/407",
language = "French",
MRtitle = "Une histoire chinoise du ``nombre {$\pi$}''",
}
@InProceedings{Kanada:1988:VMA,
author = "Yasumasa Kanada",
title = "Vectorization of multiple-precision arithmetic program
and 201,326,000 decimal digits of {$ \pi $}
calculation",
crossref = "Martin:1988:SPN",
volume = "2",
pages = "117--128",
year = "1988",
bibdate = "Sat Jul 16 16:53:44 MDT 2005",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
abstract = "More than 200 million decimal places of {$ \pi $} were
calculated using an arithmetic geometric mean formula
independently discovered by E. Salamin and R. P. Brent
in 1976. Correctness of the calculation was verified
through Borwein's quartic convergent formula developed
in 1983. The computation took CPU times of 5 hours 57
minutes for the main calculation and 7 hours 30 minutes
for the verification calculation on the HITAC S-820
model 80 supercomputer with 256 MB of main memory and 3
GB of high speed semiconductor storage, Extended
Storage, to shorten I/O time.\par
Computation was completed in 27th of January 1988. At
that day two programs generated values up to $ 3 \times
2^{26} $, about 201 million. The two results agreed
except for the last 21 digits. These results also agree
with the 133,554,000 places of calculation of $ \pi $
which was done by the author in January 1987. Compare
to the record in 1987, 50\% more decimal digits were
calculated with about $ 1 / 6 $ of CPU
time.\par
Computation was performed with real arithmetic based
vectorized Fast Fourier Transform (FFT) multiplier and
newly vectorized multiple-precision add, subtract and
(single word) constant multiplication programs.
Vectorizations for the later cases were realized
through first order linear recurrence vector
instruction on the S-820. Details of the computation
and statistical tests on the first 200 million digits
of $ \pi - 3 $ are reported.",
acknowledgement = ack-nhfb,
classification = "C4190 (Other numerical methods); C7310
(Mathematics)",
corpsource = "Comput. Centre, Tokyo Univ., Japan",
keywords = "arithmetic geometric mean formula; Borwein's quartic
convergent formula; fast Fourier transform; fast
Fourier transforms; first order linear recurrence
vector instruction; HITAC S-820 model 80 supercomputer;
mathematics computing; multiple-precision arithmetic
program; multiplier; parallel processing; pi
calculation; S-820; vectorization",
sponsororg = "IEEE; ACM SIGARCH",
treatment = "P Practical",
}
@Article{Bailey:1989:NRR,
author = "David H. Bailey and Helaman R. P. Ferguson",
title = "Numerical results on relations between fundamental
constants using a new algorithm",
journal = j-MATH-COMPUT,
volume = "53",
number = "188",
pages = "649--656",
month = oct,
year = "1989",
CODEN = "MCMPAF",
ISSN = "0025-5718 (print), 1088-6842 (electronic)",
ISSN-L = "0025-5718",
MRclass = "11Y16 (68Q25)",
MRnumber = "90e:11191",
MRreviewer = "Brigitte Vall{\'e}e",
bibdate = "Tue Oct 13 08:06:19 MDT 1998",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib; JSTOR
database",
acknowledgement = ack-nhfb,
classcodes = "C1160 (Combinatorial mathematics)",
corpsource = "NASA Ames Res. Center, Moffett Field, CA, USA",
fjournal = "Mathematics of Computation",
journal-URL = "http://www.ams.org/mcom/",
keywords = "algebraic polynomials; algorithm; bounds; fundamental
constants; integer relation; mathematical constants;
multiprecision arithmetic; number theory; numbers;
numerical; real; relation-finding algorithm; relations;
vector",
ORCID-numbers = "Bailey, David H./0000-0002-7574-8342",
treatment = "T Theoretical or Mathematical",
}
@Article{Borwein:1989:RME,
author = "J. M. Borwein and P. B. Borwein and D. H. Bailey",
title = "{Ramanujan}, modular equations, and approximations to
$ \pi $ or how to compute one billion digits of $ \pi
$",
journal = j-AMER-MATH-MONTHLY,
volume = "96",
number = "3",
pages = "201--219",
month = mar,
year = "1989",
CODEN = "AMMYAE",
DOI = "https://doi.org/10.2307/2325206",
ISSN = "0002-9890 (print), 1930-0972 (electronic)",
ISSN-L = "0002-9890",
MRclass = "11Y60 (01A60 11F03 33A25)",
MRnumber = "991866 (90d:11143)",
MRreviewer = "Herman J. J. te Riele",
bibdate = "Fri Nov 8 18:01:57 MST 2002",
bibsource = "ACM Computing Archive CD-ROM database (1991);
https://www.math.utah.edu/pub/tex/bib/pi.bib",
acknowledgement = ack-nhfb,
affiliation = "Dalhousie Univ., Halifax; Dalhousie Univ., Halifax",
author-dates = "Jonathan Michael Borwein (20 May 1951--2 August
2016)",
bibno = "65243",
catcode = "I.1.2; G.1.2; G.1.8; G.1.4; I.1.3; F.2.1; F.2.1",
CRclass = "I.1.2 Algorithms; I.1.2 Algebraic algorithms; G.1.2
Approximation; G.1.2 Elementary function approximation;
G.1.8 Partial Differential Equations; G.1.8 Elliptic
equations; G.1.4 Quadrature and Numerical
Differentiation; G.1.4 Multiple quadrature; I.1.3
Languages and Systems; F.2.1 Numerical Algorithms and
Problems; F.2.1 Computation of transforms; F.2.1
Numerical Algorithms and Problems; F.2.1
Number-theoretic computations",
descriptor = "Computing Methodologies, ALGEBRAIC MANIPULATION,
Algorithms, Algebraic algorithms; Mathematics of
Computing, NUMERICAL ANALYSIS, Approximation,
Elementary function approximation; Mathematics of
Computing, NUMERICAL ANALYSIS, Partial Differential
Equations, Elliptic equations; Mathematics of
Computing, NUMERICAL ANALYSIS, Quadrature and Numerical
Differentiation, Multiple quadrature; Computing
Methodologies, ALGEBRAIC MANIPULATION, Languages and
Systems; Theory of Computation, ANALYSIS OF ALGORITHMS
AND PROBLEM COMPLEXITY, Numerical Algorithms and
Problems, Computation of transforms; Theory of
Computation, ANALYSIS OF ALGORITHMS AND PROBLEM
COMPLEXITY, Numerical Algorithms and Problems,
Number-theoretic computations",
fjournal = "American Mathematical Monthly",
genterm = "algorithms; theory",
guideno = "1989-03459",
journal-URL = "https://www.jstor.org/journals/00029890.htm",
journalabbrev = "Am. Math. Monthly",
jrldate = "March 1989",
ORCID-numbers = "Bailey, David H./0000-0002-7574-8342; Borwein,
Jonathan/0000-0002-1263-0646",
subject = "F. Theory of Computation; F.2 ANALYSIS OF ALGORITHMS
AND PROBLEM COMPLEXITY; G. Mathematics of Computing;
G.1 NUMERICAL ANALYSIS; I. Computing Methodologies; I.1
ALGEBRAIC MANIPULATION",
}
@Article{Chudnovsky:1989:CCC,
author = "D. Chudnovsky and G. Chudnovsky",
title = "The computation of classical constants",
journal = j-PROC-NATL-ACAD-SCI-USA,
volume = "86",
number = "21",
pages = "8178--8182",
month = "????",
year = "1989",
CODEN = "PNASA6",
ISSN = "0027-8424 (print), 1091-6490 (electronic)",
ISSN-L = "0027-8424",
MRclass = "11Y60 (11-04 11Y35 33A99)",
MRnumber = "1021452 (90m:11206)",
MRreviewer = "F. Beukers",
bibdate = "Tue Apr 26 09:45:11 2011",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
URL = "http://www.pnas.org/content/86/21/8178.full.pdf+html",
abstract = "Hypergeometric representations of classical constants
and efficient algorithms for their calculation are
discussed. Particular attention is devoted to
algorithms for computing $ \pi $.",
acknowledgement = ack-nhfb,
fjournal = "Proceedings of the {National Academy of Sciences of
the United States of America}",
journal-URL = "http://www.pnas.org/search",
mathscinetremark = "In this very interesting paper the authors make a
large number of valuable comments on mathematics and
algorithmics in connection with their computation of
$\pi$ up to one billion digits. They give a short
history of the computation of $\pi$ and some remarks on
the evaluation of values of the hypergeometric
functions. They explain how the Legendre relations for
elliptic curves with complex multiplication give rise
to Ramanujan's series which are now used to compute
$\pi$. Finally, some remarks on computer
implementations are made",
}
@Article{Haastad:1989:PTA,
author = "J. H{\aa}stad and B. Just and J. C. Lagarias and C.-P.
Schnorr",
title = "Polynomial time algorithms for finding integer
relations among real numbers",
journal = j-SIAM-J-COMPUT,
volume = "18",
number = "5",
pages = "859--881",
month = oct,
year = "1989",
CODEN = "SMJCAT",
ISSN = "0097-5397 (print), 1095-7111 (electronic)",
ISSN-L = "0097-5397",
MRclass = "11Y65 (11J13 11Y16 68Q20 68Q25)",
MRnumber = "90g:11171",
MRreviewer = "W. W. Adams",
bibdate = "Mon Nov 29 11:01:23 MST 2010",
bibsource = "http://epubs.siam.org/sam-bin/dbq/toclist/SICOMP/18/5;
https://www.math.utah.edu/pub/tex/bib/pi.bib",
note = "See also earlier version in \cite{Haastad:1985:PTA}.",
acknowledgement = ack-nhfb,
fjournal = "SIAM Journal on Computing",
journal-URL = "http://epubs.siam.org/sicomp",
}
@Article{Jochi:1989:CMA,
author = "Shigeru Jochi",
title = "{Zu Chongzhi's Da Ming Almanac} and computation of $
\pi $",
journal = "J. Beijing Norm. Univ., Nat. Sci.",
volume = "1989",
number = "4",
pages = "85--89",
year = "1989",
bibdate = "Mon Apr 25 17:58:28 2011",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
ZMnumber = "0714.01002",
abstract = "After briefly describing Zu Chongzhi's contribution of
the history of Chinese astronomy, this paper deals with
Zu's famous contribution in mathematics, namely, the
discovery of the ratio $ 355 / 113 $ which is correct
to the seventh decimal place as the approximate value
for $ \pi $. The motivation of this ratio is sought to
Liu Hui's ratio $ 3927 / 1250 $. When the latter is
expressed in continuous fraction down to the third term
by so called Euclid's algorithm of division, the former
is obtained. To the reviewer it is interesting that
these two ratios are also found in Sanskrit texts and
have the similar relation as in China. See T. Hayashi,
T. Kusuba and M. Yano [Hist. Sci. 37, 1--16 (1989; Zbl
0677.01003)].",
acknowledgement = ack-nhfb,
classmath = "*01A27 (Japanese mathematics)",
fjournal = "J. Beijing Norm. Univ., Nat. Sci.",
keywords = "Chinese mathematics. continuous fraction; Euclid's
algorithm; Liu Hui; value of $\pi $",
language = "Chinese with English summary",
reviewer = "M. Yano",
}
@Article{Tee:1989:NBA,
author = "Garry J. Tee",
title = "A note on {Bechmann}'s approximate construction of $
\pi $, suggested by a deleted sketch in {Villard de
Honnecourt}'s manuscript",
journal = j-BRITISH-J-HIST-SCI,
volume = "22",
number = "2",
pages = "241--242",
month = jul,
year = "1989",
CODEN = "BJHSAT",
DOI = "https://doi.org/10.1017/S0007087400026017",
ISSN = "0007-0874 (print), 1474-001X (electronic)",
ISSN-L = "0007-0874",
MRclass = "01A35 (Mathematics in the medieval) 00A99
(Miscellaneous topics in general mathematics)",
MRnumber = "1046122 (91a:01014)",
MRreviewer = "H. L. L. Busard",
bibdate = "Thu Sep 23 07:34:43 MDT 2010",
bibsource = "http://journals.cambridge.org/action/displayJournal?jid=BJH;
https://www.math.utah.edu/pub/tex/bib/pi.bib; MathSciNet
database",
URL = "http://www.jstor.org/stable/4026662",
ZMnumber = "0682.01020",
abstract = "In this note, the author points out that a ruler and
compass construction presented in an earlier article
[{\em R. Bechmann}, ``About some technical sketches of
Villard de Honnecourt's manuscript. New light on
deleted diagrams: an unknown drawing'', Br. J. Hist.
Sci. 21, 341-361 (1988)] and inspired by a deleted
sketch in Villard de Honnecourt's sketchbook is not,
and cannot be, an exact construction of the circular
perimeter; but that it yields an excellent
approximation ($ \approx 3.1416408 R$).",
acknowledgement = ack-nhfb,
fjournal = "British Journal for the History of Science",
journal-URL = "http://journals.cambridge.org/action/displayJournal?jid=BJH",
keywords = "circle squaring",
xxnumber = "2(73)",
ZMreviewer = "J. H{\o}yrup",
}
@Article{Bailey:1990:FEH,
author = "David H. Bailey",
title = "{FFTs} in External or Hierarchical Memory",
journal = j-J-SUPERCOMPUTING,
volume = "4",
number = "1",
pages = "23--35",
month = mar,
year = "1990",
CODEN = "JOSUED",
DOI = "https://doi.org/10.1007/BF00162341",
ISSN = "0920-8542 (print), 1573-0484 (electronic)",
ISSN-L = "0920-8542",
bibdate = "Wed Jul 6 11:13:01 MDT 2005",
bibsource = "ftp://ftp.ira.uka.de/pub/Parallel/JOURNAL.SUPER.bib;
http://springerlink.metapress.com/openurl.asp?genre=issue&issn=0920-8542&volume=4&issue=1;
https://www.math.utah.edu/pub/tex/bib/jsuper.bib;
https://www.math.utah.edu/pub/tex/bib/pi.bib",
URL = "http://www.springerlink.com/openurl.asp?genre=article&issn=0920-8542&volume=4&issue=1&spage=23",
acknowledgement = ack-nhfb,
affiliation = "Numerical Aerodynamic Simulation Syst. Div., NASA Ames
Res. Center, Moffett Field, CA, USA",
classification = "C4190 (Other numerical methods); C5310 (Storage
system design); C5440 (Multiprocessor systems and
techniques); C6120 (File organisation)",
corpsource = "Numerical Aerodynamic Simulation Syst. Div., NASA Ames
Res. Center, Moffett Field, CA, USA",
fjournal = "The Journal of Supercomputing",
journal-URL = "http://link.springer.com/journal/11227",
keywords = "2 GFLOPS; advanced techniques; Cray library FFT
routines; Cray supercomputers; CRAY X-MP; CRAY Y-MP
systems; CRAY-2; data structures; external data set;
external storage; fast Fourier transforms; FFT
algorithms; hierarchical memory; large one-dimensional
fast Fourier transforms; long vector transfers; main
memory; memory architecture; ordered FFT; parallel
algorithms; parallel computation; parallel computers;
parallel machines; scratch space; storage management;
unit stride",
ORCID-numbers = "Bailey, David H./0000-0002-7574-8342",
remark = "The work in this paper originated in work on computing
$ \pi $ for testing of supercomputer circuitry.",
treatment = "P Practical",
}
@Article{Desbrow:1990:NI,
author = "D. Desbrow",
title = "Notes: On the Irrationality of $ \pi^2 $",
journal = j-AMER-MATH-MONTHLY,
volume = "97",
number = "10",
pages = "903--906",
month = dec,
year = "1990",
CODEN = "AMMYAE",
ISSN = "0002-9890 (print), 1930-0972 (electronic)",
ISSN-L = "0002-9890",
MRclass = "11J72",
MRnumber = "91j:11055",
MRreviewer = "Jaroslav Han{\u{c}}l",
bibdate = "Mon Jun 28 12:36:11 MDT 1999",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib; JSTOR
database",
acknowledgement = ack-nhfb,
fjournal = "American Mathematical Monthly",
journal-URL = "https://www.jstor.org/journals/00029890.htm",
}
@Article{Johnson:1990:SDC,
author = "Bruce R. Johnson and David J. Leeming",
title = "A study of the digits of $ \pi $, $e$, and certain
other irrational numbers",
journal = j-SANKHYA-B,
volume = "52",
number = "2",
pages = "183--189",
month = "????",
year = "1990",
CODEN = "SANBBV",
ISSN = "0581-5738",
bibdate = "Fri Jul 01 10:43:38 2011",
bibsource = "http://sankhya.isical.ac.in/index.html;
https://www.math.utah.edu/pub/tex/bib/pi.bib",
abstract = "The first 100,000 digits in the decimal expansions of
$ \pi $, $e$, $ \sqrt {2}$, $ \sqrt {3}$, $ \sqrt {5}$,
$ \sqrt {7}$, $ \sqrt {11}$ and $ \sqrt {13}$ were
investigated for properties of randomness. Using a
measure of randomness based on several different runs
statistics, the decimal expansions of these irrational
numbers behaved very much like random sequences when
compared to the outputs of two popular random number
generators. Also, for a better understanding of power,
the measure of randomness was evaluated for several
different kinds of nonrandom digit sequences.",
acknowledgement = ack-nhfb,
fjournal = "Sankhy{\=a} (Indian Journal of Statistics), Series B.
Methodological",
remark = "The authors report statistics for the randomness of
the first 100,000 digits of $ \pi $, $e$, $ \sqrt {2}$,
$ \sqrt {3}$, $ \sqrt {5}$, $ \sqrt {7}$, $ \sqrt
{11}$, and $ \sqrt {13}$, and show that the digits of $
\pi $ and $ \sqrt {7}$ appear to be more random than
those from \texttt{urand()} and \texttt{c05dyf()}.",
xxnote = "The journal Web site does not have an online form of
this article.",
}
@TechReport{Bailey:1991:PTN,
author = "D. H. Bailey and H. R. P. Ferguson",
title = "A polynomial time, numerically stable integer relation
algorithm",
type = "Report",
number = "SRC-TR-92-066",
institution = "Supercomputing Research Center",
address = "????",
pages = "1--14",
day = "16",
month = dec,
year = "1991",
bibdate = "Tue Apr 26 17:03:43 2011",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
note = "Also issued as RNR Technical Report RNR-91-032 (16
December 1991; 14 July 1992), NASA Ames Research
Center, MS T045-1, Moffett Field, CA 94035-1000.",
acknowledgement = ack-nhfb,
keywords = "precursor of PSLQ algorithm",
ORCID-numbers = "Bailey, David H./0000-0002-7574-8342",
}
@Article{Gillman:1991:TML,
author = "Leonard Gillman",
title = "The Teaching of Mathematics: $ \pi $ and the Limit of
$ (\sin \alpha) / \alpha $",
journal = j-AMER-MATH-MONTHLY,
volume = "98",
number = "4",
pages = "346--349",
month = apr,
year = "1991",
CODEN = "AMMYAE",
ISSN = "0002-9890 (print), 1930-0972 (electronic)",
ISSN-L = "0002-9890",
bibdate = "Mon Jun 28 12:36:19 MDT 1999",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib; JSTOR
database",
acknowledgement = ack-nhfb,
fjournal = "American Mathematical Monthly",
journal-URL = "https://www.jstor.org/journals/00029890.htm",
}
@Article{Lynch:1990:DHO,
author = "R. Lynch and H. A. Mavromatis",
title = "{$N$}-dimensional harmonic oscillator yields monotonic
series for the mathematical constant $ \pi $",
journal = j-J-COMPUT-APPL-MATH,
volume = "30",
number = "2",
pages = "127--137",
day = "28",
month = may,
year = "1990",
CODEN = "JCAMDI",
ISSN = "0377-0427 (print), 1879-1778 (electronic)",
ISSN-L = "0377-0427",
bibdate = "Sat Feb 25 12:20:45 MST 2017",
bibsource = "https://www.math.utah.edu/pub/tex/bib/jcomputapplmath1990.bib;
https://www.math.utah.edu/pub/tex/bib/pi.bib",
URL = "http://www.sciencedirect.com/science/article/pii/037704279090021Q",
acknowledgement = ack-nhfb,
fjournal = "Journal of Computational and Applied Mathematics",
journal-URL = "http://www.sciencedirect.com/science/journal/03770427/",
}
@Book{Schroeder:1991:FCP,
author = "Manfred Schroeder",
title = "Fractals, Chaos, Power Laws",
publisher = pub-W-H-FREEMAN,
address = pub-W-H-FREEMAN:adr,
pages = "xviii + 429",
year = "1991",
ISBN = "0-671-74217-5, 0-7167-2136-8, 0-7167-2357-3",
ISBN-13 = "978-0-671-74217-1, 978-0-7167-2136-9,
978-0-7167-2357-8",
LCCN = "QD921 .S3 1990",
bibdate = "Wed Dec 15 10:41:35 1993",
bibsource = "https://www.math.utah.edu/pub/bibnet/authors/h/heisenberg-werner.bib;
https://www.math.utah.edu/pub/bibnet/authors/m/mandelbrot-benoit.bib;
https://www.math.utah.edu/pub/bibnet/authors/s/shannon-claude-elwood.bib;
https://www.math.utah.edu/pub/tex/bib/benfords-law.bib;
https://www.math.utah.edu/pub/tex/bib/fibquart.bib;
https://www.math.utah.edu/pub/tex/bib/master.bib;
https://www.math.utah.edu/pub/tex/bib/pi.bib",
abstract = "Reveals the extraordinary dimensions of new
mathematical insights about the nature of physical
reality; explores the powerful applications of these
symmetry concepts in physics, chemistry, music, and the
visual arts. Includes such areas as deterministic chaos
and strange attractors, iterated mappings, nonlinear
dynamics, Cayley trees, cellular automata, random
fractals and related topics.",
acknowledgement = ack-nhfb,
shorttableofcontents = "Introduction \\
Similarity and dissimilarity \\
Self-similarity --- discrete, continuous, strict, and
otherwise \\
Power laws: endless sources of self-similarity \\
Noises: white, pink, brown, and black \\
Brownian motion, gambling losses, and intergalactic
voids: random fractals par excellence \\
Cantor sets: self-similarity and arithmetic dust \\
Fractals in higher dimensions and a digital sundial \\
Multifractals: intimately intertwined fractals \\
Some practical fractals and their measurement \\
Iteration, strange mappings, and a billion digits for
$\pi$ \\
A self-similar sequence, the logistic parabola, and
symbolic dynamics \\
A forbidden symmetry, Fibonacci's rabbits, and a new
state of matter \\
Periodic and quasiperiodic structures in space --- the
route to spatial chaos \\
Percolation: from forest fires to epidemics \\
Phase transitions and renormalization \\
Cellular automata",
tableofcontents = "Preface / xiii \\
1: Introduction / 1 \\
Einstein, Pythagoras, and Simple Similarity / 3 \\
A Self-Similar Array of Self-Preserving Queens / 4 \\
A Self-Similar Snowflake / 7 \\
A New Dimension for Fractals / 9 \\
A Self-Similar Tiling and a ``Non-Euclidean'' Paradox /
13 \\
At the Gates of Cantor's Paradise / 15 \\
The Sierpinski Gasket / 17 \\
Sir Pinski's Game and Deterministic Chaos / 20 \\
Three Bodies Cause Chaos / 25 \\
Strange Attractors, Their Basins, and a Chaos Game 2 /
7 \\
Percolating Random Fractals / 30 \\
Power Laws: from Alvarez to Zipf / 33 \\
Newton's Iteration and How to Abolish Two-Nation
Boundaries / 38 \\
Could Minkowski Hear the Shape of a Drum? / 40 \\
Discrete Self-Similarity: Creases and Center Folds / 45
\\
Golden and Silver Means and Hyperbolic Chaos / 49 \\
Winning at Fibonacci Nim / 53 \\
Self-Similar Sequences from Square Lattices / 55 \\
John Horton Conway's ``Death Bet'' / 57 \\
2: Similarity and Dissimilarity / 61 \\
More Than One Scale / 61 \\
To Scale or Not to Scale: A Bit of Biology and
Astrophysics / 63 \\
Similarity in Physics: Some Astounding Consequences /
66 \\
Similarity in Concert Halls, Microwaves, and
Hydrodynamics / 68 \\
Scaling in Psychology / 70 \\
Acousticians, Alchemy, and Concert Halls / 72 \\
Preference and Dissimilarity: Concert Halls Revisited /
74 \\
3: Self-Similarity --- Discrete, Continuous, Strict,
and Otherwise / 81 \\
The Logarithmic Spiral, Cutting Knives, and Wideband
Antennas / 89 \\
Some Simple Cases of Self-Similarity / 93 \\
Weierstrass Functions and a Musical Paradox / 96 \\
More Self-Similarity in Music: The Tempered Scales of
Bach / 99 \\
The Excellent Relations between the Primes 3, 5, and 7
/ 102 \\
4: Power Laws: Endless Sources of Self-Similarity / 103
\\
The Sizes of Cities and Meteorites / 103 \\
A Fifth Force of Attraction / 105 \\
Free of Natural Scales / 107 \\
Bach Composing on All Scales / 107 \\
Birkhoff's Aesthetic Theory / 109 \\
Heisenberg's Hyperbolic Uncertainty Principle / 112 \\
Fractional Exponents / 115 \\
The Peculiar Distribution of the First Digit / 116 \\
The Diameter Exponents of Trees, Rivers, Arteries, and
Lungs / 117 \\
5: Noises: White, Pink, Brown, and Black / 121 \\
Pink Noise / 122 \\
Self-Similar Trends on the Stock Market / 126 \\
Black Noises and Nile Floods / 129 \\
Warning: World Warming / 131 \\
Fractional Integration: A Modem Tool / 131 \\
Brownian Mountains / 133 \\
Radon Transform and Computer Tomography / 134 \\
Fresh and Tired Mountains / 135 \\
6: Brownian Motion, Gambling Losses, and Intergalactic
Voids: Random Fractals Par Excellence / 139 \\
The Brownian Beast Tamed / 140 \\
Brownian Motion as a Fractal / 141 \\
How Many Molecules? / 143 \\
The Spectrum of Brownian Motion / 144 \\
The Gambler's Ruin, Random Walks, and Information
Theory / 145 \\
Counterintuition Runs Rampant in Random Runs / 146 \\
More Food for Fair Thought / 147 \\
The St. Petersburg Paradox / 148 \\
Shannon's Outguessing Machine / 149 \\
The Classical Mechanics of Roulette and Shannon's
Channel Capacity / 150 \\
The Clustering of Poverty and Galaxies / 152 \\
Levy Flights through the Universe / 155 \\
Paradoxes from Probabilistic Power Laws / 155 \\
Invariant Distributions: Gauss, Cauchy, and Beyond /
157 \\
7: Cantor Sets: Self-Similarity and Arithmetic Dust /
161 \\
A Comer of Cantor's Paradise / 161 \\
Cantor Sets as Invariant Sets / 165 \\
Symbolic Dynamics and Deterministic Chaos / 166 \\
Devil's Staircases and a Pinball Machine / 167 \\
Mode Locking in Swings and Clocks / 171 \\
The Frustrated Manhattan Pedestrian / 172 \\
Arnold Tongues 17 / 4 \\
8: Fractals in Higher Dimensions and a Digital Sundial
/ 177 \\
Cartesian Products of Cantor Sets / 177 \\
A Leaky Gasket, Soft Sponges, and Swiss Cheeses / 178
\\
A Cantor-Set Sundial / 181 \\
Fat Fractals / 183 \\
9: Multifractals: Intimately Intertwined Fractals / 187
\\
The Distributions of People and Ore / 187 \\
Self-Affine Fractals without Holes / 190 \\
The Multifractal Spectrum: Turbulence and
Diffusion-Limited Aggregation / 193 \\
Viscous Fingering / 199 \\
Multifractals on Fractals / 200 \\
Fractal Dimensions from Generalized Entropies / 203 \\
The Relation between the Multifractal Spectrum
$f(\alpha)$ and the Mass Exponents $(q)$ / 205 \\
Strange Attractors as Multifractals / 206 \\
A Greedy Algorithm for Unfavorable Odds / 207 \\
10: Some Practical Fractals and Their Measurement / 211
\\
Dimensions from Box Counting / 213 \\
The Mass Dimension / 215 \\
The Correlation Dimension / 220 \\
Infinitely Many Dimensions / 220 \\
The Determination of Fractal Dimensions from Time
Series / 223 \\
Abstract Concrete / 224 \\
Fractal Interfaces Enforce Fractional Frequency
Exponents / 225 \\
The Fractal Dimensions of Fracture Surfaces / 230 \\
The Fractal Shapes of Clouds and Rain Areas / 231 \\
Cluster Agglomeration / 232 \\
Diffraction from Fractals / 233 \\
11: Iteration, Strange Mappings, and a Billion Digits
for Pi / 237 \\
Looking for Zeros and Encountering Chaos / 239 \\
The Strange Sets of Julia / 243 \\
A Multifractal Julia Set / 245 \\
The Beauty of Broken Linear Relationships / 249 \\
The Baker's Transformation and Digital Musical Chairs /
251 \\
Arnol'd's Cat Map / 253 \\
A Billion Digits for $\pi$ / 257 \\
Bushes and Flowers from Iterations / 259 \\
12: A Self-Similar Sequence, the Logistic Parabola, and
Symbolic Dynamics / 263 \\
Self-Similarity from the Integers / 264 \\
The Logistic Parabola and Period Doubling / 268 \\
Self-Similarity in the Logistic Parabola / 272 \\
The Scaling of the Growth Parameter / 274 \\
Self-Similar Symbolic Dynamics / 277 \\
Periodic Windows Embedded in Chaos / 279 \\
The Parenting of New Orbits / 282 \\
The Calculation of the Growth Parameters for Different
Orbits / 286 \\
Tangent Bifurcations, Intermittency, and I/f Noise /
289 \\
A Case of Complete Chaos / 291 \\
The Mandelbrot Set / 295 \\
The Julia Sets of the Complex Quadratic Map / 297 \\
13: A Forbidden Symmetry, Fibonacci's Rabbits, and a
New State of Matter / 301 \\
The Forbidden Fivefold Symmetry / 301 \\
Long-Range Order from Neighborly Interactions / 304 \\
Generation of the Rabbit Sequence from the Fibonacci
Number System / 307 \\
The Self-Similar Spectrum of the Rabbit Sequence / 308
\\
Self-Similarity in the Rabbit Sequence / 310 \\
A One-Dimensional Quasiperiodic Lattice / 310 \\
Self-Similarity from Projections / 311 \\
More Forbidden Symmetries / 315 \\
14: Periodic and Quasiperiodic Structures in Space ---
The route to Spatial Chaos / 319 \\
Periodicity and Quasiperiodicity in Space / 320 \\
The Devil's Staircase for Ising Spins / 321 \\
Quasiperiodic Spatial Distributions / 322 \\
Beatty Sequence Spins / 325 \\
The Scaling Laws for Quasiperiodic Spins / 329 \\
Self-Similar Winding Numbers / 330 \\
Circle Maps and Arnold Tongues / 331 \\
Mediants, Farey Sequences, and the Farey Tree / 334 \\
The Golden-Mean Route to Chaos / 340 \\
15: Percolation: From Forest Fires to Epidemics / 345
\\
Critical Conflagration on a Square Lattice / 346 \\
Universality / 350 \\
The Critical Density / 353 \\
The Fractal Perimeters of Percolation / 353 \\
Finite-Size Scaling / 354 \\
16: Phase Transitions and Renormalization / 357 \\
A First-Order Markov Process / 357 \\
Self-Similar and Non-Self-Similar Markov Processes /
358 \\
The Scaling of Markov Output's / 360 \\
Renormalization and Hierarchical Lattices / 362 \\
The Percolation Threshold of the Bethe Lattice / 363
\\
A Simple Renormalization / 367 \\
17: Cellular Automata / 371 \\
The Game of Life / 373 \\
Cellular Growth and Decay / 375 \\
Biological Pattern Formation / 382 \\
Self-Similarity from a Cellular Automaton / 383 \\
A Catalytic Converter as a Cellular Automaton / 386 \\
Pascal's Triangle Modulo $N$ / 387 \\
Bak's Self-Organized Critical Sandpiles / 389 \\
Appendix / 391 \\
References / 395 \\
Author Index / 411 \\
Subject Index / 417",
}
@Article{Tweddle:1991:JMR,
author = "Ian Tweddle",
title = "{John Machin} and {Robert Simson} on inverse-tangent
series for $ \pi $",
journal = j-ARCH-HIST-EXACT-SCI,
volume = "42",
number = "1",
pages = "1--14",
month = mar,
year = "1991",
CODEN = "AHESAN",
DOI = "https://doi.org/10.1007/BF00384331",
ISSN = "0003-9519 (print), 1432-0657 (electronic)",
ISSN-L = "0003-9519",
MRclass = "01A50",
MRnumber = "1111103 (92h:01026)",
MRreviewer = "P. Bockstaele",
bibdate = "Fri Feb 4 21:50:28 MST 2011",
bibsource = "http://springerlink.metapress.com/openurl.asp?genre=issue&issn=0003-9519&volume=42&issue=1;
https://www.math.utah.edu/pub/tex/bib/pi.bib",
URL = "http://www.springerlink.com/openurl.asp?genre=article&issn=0003-9519&volume=42&issue=1&spage=1",
acknowledgement = ack-nhfb,
fjournal = "Archive for History of Exact Sciences",
journal-URL = "http://link.springer.com/journal/407",
MRtitle = "{John Machin} and {Robert Simson} on inverse-tangent
series for {$\pi$}",
}
@Book{Barrow:1992:PSC,
author = "John D. Barrow",
title = "Pi in the Sky: Counting, Thinking, and Being",
publisher = pub-CLARENDON,
address = pub-CLARENDON:adr,
pages = "ix + 317",
year = "1992",
ISBN = "0-19-853956-8",
ISBN-13 = "978-0-19-853956-8",
LCCN = "QA36 .B37 1992",
bibdate = "Sat Dec 17 14:44:47 MST 2005",
bibsource = "https://www.math.utah.edu/pub/tex/bib/master.bib;
https://www.math.utah.edu/pub/tex/bib/pi.bib;
z3950.loc.gov:7090/Voyager",
price = "US\$30.00 (Oxford Univ. Press)",
abstract = "John D. Barrow's \booktitle{Pi in the Sky} is a
profound --- and profoundly different --- exploration
of the world of mathematics: where it comes from, what
it is, and where it's going to take us if we follow it
to the limit in our search for the ultimate meaning of
the universe. Barrow begins by investigating whether
math is a purely human invention inspired by our
practical needs. Or is it something inherent in nature
waiting to be discovered? In answering these questions,
Barrow provides a bridge between the usually
irreconcilable worlds of mathematics and theology.
Along the way, he treats us to a history of counting
all over the world, from Egyptian hieroglyphics to
logical friction, from number mysticism to Marxist
mathematics. And he introduces us to a host of peculiar
individuals who have thought some of the deepest and
strangest thoughts that human minds have ever thought,
from Lao-Tse to Robert Pirsig, Charles Darwin, and
Umberto Eco. Barrow thus provides the historical
framework and the intellectual tools necessary to an
understanding of some of today's weightiest
mathematical concepts.",
acknowledgement = ack-nhfb,
libnote = "Not in my library.",
subject = "Mathematics",
tableofcontents = "1: From mystery to history \\
A mystery within an enigma \\
Illusions of certainty \\
The secret society \\
Non-Euclideanism \\
Logics \\
To Be or Not To Be \\
The Rashomon effect \\
The analogy that never breaks down? \\
Tinkling symbols \\
Thinking about thinking \\
2: The Counter Culture \\
By the pricking of my thumbs \\
The bare bones of history \\
Creation or evolution \\
The ordinals versus the cardinals \\
Counting without counting \\
Fingers and toes \\
Baser methods \\
Counting with base 2 \\
The neo-2 system of counting \\
Counting in fives \\
What's so special about sixty? \\
The spread of the decimal system \\
The dance of the seven veils \\
Ritual geometry \\
The place-value system and the invention of zero \\
A final accounting \\
3: With form but void \\
Numerology \\
The very opposite \\
Hilbert's scheme \\
Kurt G{\"o}del \\
More surprises \\
Thinking by numbers \\
Bourbachique math{\'e}matique \\
Arithmetic in chaos \\
Science friction \\
Mathematicians off form \\
4: The mothers of inventionism \\
Mind from matter \\
Shadowlands \\
Trap-door functions \\
Mathematical creation \\
Marxist mathematics \\
Complexity and simplicity \\
Maths as psychology \\
Pre-established mental harmony? \\
Self-discovery \\
5: Intuitionism: the immaculate construction \\
Mathematicians from outer space \\
Ramanujan \\
Intuitionism and three-valued logic \\
A very peculiar practice \\
A closer look at Brouwer \\
What is 'intuition'? The tragedy of Cantor and
Kronecker \\
Cantor and infinity \\
The comedy of Hilbert and Brouwer \\
The Four-Colour Conjecture \\
Transhuman mathematics \\
New-age mathematics \\
Paradigms \\
Computability, compressibility, and utility \\
6: Platonic heavens above and within \\
The growth of abstraction \\
Footsteps through Plato's footnotes \\
The platonic world of mathematics \\
Far away and long ago \\
The presence of the past \\
The unreasonable effectiveness of mathematics \\
Difficulties with platonic relationships \\
Seance or science? \\
Revel without a cause \\
A computer ontological argument \\
A speculative anthropic interpretation of mathematics.
\\
Maths and mysticism \\
Supernatural numbers?",
}
@InCollection{Freguglia:1992:DFP,
author = "Paolo Freguglia",
booktitle = "Contributions to the history of mathematics
({Italian}) ({Modena}, 1990)",
title = "The determination of {$ \pi $} in {Fibonacci}'s {{\it
Practica geometriae}} in a fifteenth-century
manuscript",
volume = "8",
publisher = "Accad. Naz. Sci. Lett. Arti",
address = "Modena, Italy",
pages = "75--84",
year = "1992",
MRclass = "01A35",
MRnumber = "1223787 (94c:01008)",
bibdate = "Mon Apr 25 16:27:00 2011",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
series = "Coll. Studi",
acknowledgement = ack-nhfb,
}
@Article{Lord:1992:RCG,
author = "Nick Lord",
title = "Recent calculations of $ \pi $: The {Gauss--Salamin}
algorithm",
journal = j-MATH-GAZ,
volume = "76",
number = "476",
pages = "231--242",
year = "1992",
CODEN = "MAGAAS",
DOI = "https://doi.org/10.2307/3619132",
ISSN = "0025-5572; 2056-6328/e",
MRnumber = "11Y16 11-01 33E05",
bibdate = "Thu Jun 9 15:49:28 2016",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
ZMnumber = "0805.11086",
acknowledgement = ack-nhfb,
fjournal = "The Mathematical Gazette",
journal-URL = "http://www.m-a.org.uk/jsp/index.jsp?lnk=620",
language = "English",
}
@Book{Mauron:1992:P,
author = "C. Mauron",
title = "$ \pi $ [pi]",
publisher = "Mauron and Lachat",
address = "Fribourg, Switzerland",
pages = "????",
year = "1992",
ISBN = "????",
ISBN-13 = "????",
LCCN = "????",
bibdate = "Fri Jul 01 09:57:30 2011",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
note = "Mauron computes $ \pi $ to 1,000,000 decimal digits
using independent formulas of Liebniz, Machin, and
St{\"o}rmer.",
acknowledgement = ack-nhfb,
remark = "Is this a book, or a technical report? I cannot find
it in major library catalogs.",
}
@Article{Abeles:1993:CDG,
author = "Francine F. Abeles",
title = "{Charles L. Dodgson}'s geometric approach to
arctangent relations for Pi",
journal = j-HIST-MATH,
volume = "20",
number = "2",
pages = "151--159",
month = may,
year = "1993",
CODEN = "HIMADS",
DOI = "https://doi.org/10.1006/hmat.1993.1013",
ISSN = "0315-0860 (print), 1090-249X (electronic)",
ISSN-L = "0315-0860",
bibdate = "Wed Jun 26 06:18:40 MDT 2013",
bibsource = "https://www.math.utah.edu/pub/tex/bib/histmath.bib;
https://www.math.utah.edu/pub/tex/bib/pi.bib",
URL = "http://www.sciencedirect.com/science/article/pii/S031508608371013X",
acknowledgement = ack-nhfb,
fjournal = "Historia Mathematica",
journal-URL = "http://www.sciencedirect.com/science/journal/03150860",
}
@TechReport{Arno:1993:NPT,
author = "Steve Arno and Helaman Ferguson",
title = "A new polynomial time algorithm for finding relations
among real numbers",
type = "Report",
number = "SRC-93-093",
institution = "Supercomputing Research Center",
address = "????",
pages = "1--13",
month = mar,
year = "1993",
bibdate = "Tue Apr 26 17:01:48 2011",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
acknowledgement = ack-nhfb,
keywords = "PSLQ algorithm (first publication of??)",
}
@Article{Bailey:1993:AMT,
author = "David H. Bailey",
title = "{Algorithm 719}: Multiprecision Translation and
Execution of {FORTRAN} Programs",
journal = j-TOMS,
volume = "19",
number = "3",
pages = "288--319",
month = sep,
year = "1993",
CODEN = "ACMSCU",
DOI = "https://doi.org/10.1145/155743.155767",
ISSN = "0098-3500 (print), 1557-7295 (electronic)",
ISSN-L = "0098-3500",
bibdate = "Wed Dec 13 18:37:31 1995",
bibsource = "ftp://garbo.uwasa.fi/pc/doc-soft/fpbibl18.zip;
https://www.math.utah.edu/pub/tex/bib/pi.bib",
URL = "http://www.acm.org/pubs/citations/journals/toms/1993-19-3/p288-bailey/",
abstract = "This paper describes two Fortran utilities for
multiprecision computation. The first is a package of
Fortran subroutines that perform a variety of
arithmetic operations and transcendental functions on
floating point numbers of arbitrarily high precision.
This package is in some cases over 200 times faster
than that of certain other packages that have been
developed for this purpose.\par
The second utility is a translator program, which
facilitates the conversion of ordinary Fortran programs
to use this package. By means of source directives
(special comments) in the original Fortran program, the
user declares the precision level and specifies which
variables in each subprogram are to be treated as
multiprecision. The translator program reads this
source program and outputs a program with the
appropriate multiprecision subroutine calls.\par
This translator supports multiprecision integer, real,
and complex datatypes. The required array space for
multiprecision data types is automatically allocated.
In the evaluation of computational expressions, all of
the usual conventions for operator precedence and mixed
mode operations are upheld. Furthermore, most of the
Fortran-77 intrinsics, such as ABS, MOD, NINT, COS, EXP
are supported and produce true multiprecision values.",
abstract-2 = "The author describes two Fortran utilities for
multiprecision computation. The first is a package of
Fortran subroutines that perform a variety of
arithmetic operations and transcendental functions on
floating point numbers of arbitrarily high precision.
This package is in some cases over 200 times faster
than that of certain other packages that have been
developed for this purpose. The second utility is a
translator program, which facilitates the conversion of
ordinary Fortran programs to use this package. By means
of source directives (special comments) in the original
Fortran program, the user declares the precision level
and specifies which variables in each subprogram are to
be treated as multiprecision. The translator program
reads this source program and outputs a program with
the appropriate multiprecision subroutine calls. This
translator supports multiprecision integer, real, and
complex datatypes. The required array space for
multiprecision data types is automatically allocated.
In the evaluation of computational expressions, all of
the usual conventions for operator precedence and mixed
mode operations are upheld. Furthermore, most of the
Fortran-77 intrinsics, such as ABS, MOD, NINT, COS, EXP
are supported and produce true multiprecision values.",
acknowledgement = ack-nhfb # " and " # ack-nj,
affiliation = "NASA Ames Res. Center, Moffett Field, CA, USA",
classification = "C5230 (Digital arithmetic methods); C6120 (File
organisation); C6140D (High level languages); C6150C
(Compilers, interpreters and other processors); C7310
(Mathematics)",
fjournal = "ACM Transactions on Mathematical Software",
journal-URL = "http://portal.acm.org/toc.cfm?idx=J782",
keywords = "Algorithm 719; Arithmetic operations; Array space;
Complex data types; Computational expressions; Floating
point numbers; Fortran programs; Fortran subroutines;
Fortran utilities; Fortran-77 intrinsics; Mixed mode
operations; Multiprecision computation; Multiprecision
data types; Multiprecision subroutine calls;
Multiprecision translation; Operator precedence; Source
directives; Transcendental functions; Translator
program",
ORCID-numbers = "Bailey, David H./0000-0002-7574-8342",
subject = "F.2.1 [Analysis of Algorithms and Problem Complexity]:
Numerical Algorithms and Problems; G.1.0 [Numerical
Analysis]: General; G.1.2 [Numerical Analysis];
Approximation",
thesaurus = "Data structures; Digital arithmetic; FORTRAN;
Mathematics computing; Program interpreters;
Subroutines",
}
@Book{Beckmann:1993:HP,
author = "Petr Beckmann",
title = "A history of $ \pi $",
publisher = pub-BARNES-NOBLE,
address = pub-BARNES-NOBLE:adr,
pages = "200",
year = "1993",
ISBN = "0-88029-418-3",
ISBN-13 = "978-0-88029-418-8",
LCCN = "QA484 .B4 1971",
bibdate = "Mon Mar 06 08:52:46 2000",
bibsource = "https://www.math.utah.edu/pub/tex/bib/master.bib;
https://www.math.utah.edu/pub/tex/bib/pi.bib",
note = "Reprint of the third edition of 1971.",
price = "US\$6.98",
abstract = "Documents the calculation, numerical value, and use of
the ratio from 2000 B.C. to the modern computer age,
detailing social conditions in eras when progress was
made.",
acknowledgement = ack-nhfb,
tableofcontents = "Dawn \\
The Belt \\
The Early Greeks \\
Euclid \\
The Roman pest \\
Archimedes of Syracuse \\
Dusk \\
Night \\
Awakening \\
The Digit hunters \\
The Last ARchimedians \\
Prelude to breakthrough \\
Newton \\
Euler \\
The Monte Carlo method \\
The Transcendence of [pi] \\
The Modern circle squares \\
The Computer age \\
Chronological table",
xxnote = "Fourth edition, 1977, Golem Press, Boulder, CO, ISBN
0-911762-18-3, LCCN QA484 .B4 1977, also available.",
}
@Article{Badger:1994:LLA,
author = "Lee Badger",
title = "{Lazzarini}'s Lucky Approximation of $ \pi $",
journal = j-MATH-MAG,
volume = "67",
number = "2",
pages = "83--91",
month = apr,
year = "1994",
CODEN = "MAMGA8",
DOI = "https://doi.org/10.2307/2690682",
ISSN = "0025-570X",
ISSN-L = "0025-570X",
bibdate = "Wed Oct 21 09:38:12 2015",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
URL = "http://www.jstor.org/stable/2690682;
http://www.maa.org/programs/maa-awards/writing-awards/lazzarinis-lucky-approximation-of-pi",
acknowledgement = ack-nhfb,
fjournal = "Mathematics Magazine",
journal-URL = "http://www.maa.org/pubs/mathmag.html",
keywords = "Buffon needle approximation of $\pi$",
remark = "The author writes: ``I will \ldots{} virtually rule
out any possibility that Lazzarini performed a valid
experiment [of Buffon needle-casting].''",
}
@Article{Bailey:1994:EEE,
author = "David H. Bailey and Jonathan M. Borwein and Roland
Girgensohn",
title = "Experimental Evaluation of {Euler} Sums",
journal = j-EXP-MATH,
volume = "3",
number = "1",
pages = "17--30",
month = "????",
year = "1994",
ISSN = "1058-6458 (print), 1944-950X (electronic)",
ISSN-L = "1058-6458",
MRnumber = "MR 96e:11168",
bibdate = "Mon Apr 25 18:38:56 2011",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
URL = "http://www.ams.org/mathscinet-getitem?mr=96e:11168",
acknowledgement = ack-nhfb,
author-dates = "Jonathan Michael Borwein (20 May 1951--2 August
2016)",
fjournal = "Experimental Mathematics",
journal-URL = "http://www.tandfonline.com/loi/uexm20",
ORCID-numbers = "Bailey, David H./0000-0002-7574-8342; Borwein,
Jonathan/0000-0002-1263-0646",
}
@Book{Clawson:1994:MTE,
author = "Calvin C. Clawson",
title = "The Mathematical Traveler: Exploring the Grand History
of Numbers",
publisher = pub-PLENUM,
address = pub-PLENUM:adr,
pages = "x + 307",
year = "1994",
ISBN = "0-306-44645-6",
ISBN-13 = "978-0-306-44645-0",
LCCN = "QA141 .C52 1994",
bibdate = "Wed Dec 31 11:51:02 1997",
bibsource = "https://www.math.utah.edu/pub/tex/bib/master.bib;
https://www.math.utah.edu/pub/tex/bib/pi.bib",
price = "US\$25.95",
abstract = "The story of numbers is a rich, sweeping history that
shows how our mathematical achievements contributed to
the greatest innovations of civilization. Calvin
Clawson, acclaimed author of \booktitle{Conquering Math
Phobia}, weaves a story of numbers that spans thousands
of years. As Clawson so clearly shows, numbers are not
only an intrinsic and essential thread in our modern
lives, but have always been an integral part of the
human psyche --- knit into the very fabric of our
identity as humans. Clawson travels back through time
to the roots of the history of numbers. In exploring
early human fascination with numbers, he unearths the
clay beads, knotted ropes, and tablets used by our
ancestors as counting tools. He then investigates how
numeric symbols and concepts developed uniquely and
independently in Meso-America, China, and Egypt. As he
persuasively argues, the mathematical concepts that
arose and flourished in the ancient world enabled the
creation of architectural masterpieces as well as the
establishment of vast trade networks. Continuing the
journey, Clawson brings us to the elegant logic of
numbers that soon came to distinguish itself as a
discipline and the language of science. From the
concepts of infinity contemplated by the Greeks to the
complex numbers that are indispensable to scientists on
the cutting edge of research today, Clawson breathes
life and meaning into the history of great mathematical
mysteries and problems. In this spirit of inquiry, he
explores, in their times and places, the discovery of
numbers that lie outside the province of counting,
including irrational numbers, transcendentals, complex
numbers, and the enormous transfinite numbers. The
personalities and the creative feats surrounding each
mathematical invention come alive vividly in Clawson's
lucid prose. In this work of breathtaking scope,
Clawson guides us through the wonders of numbers and
illustrates their monumental impact on civilization.",
acknowledgement = ack-nhfb,
subject = "Numeration; Counting; Counting; Numeration;
Getaltheorie; Geschichte; Zahlentheorie",
tableofcontents = "Acknowledgments / vii \\
Introduction / 1 \\
1: How Do We Count? / 5 \\
2: Early Counting / 19 \\
3: Counting in Other Species: How Smart Are They? / 37
\\
4: Ancient Numbers / 49 \\
5: Chinese and New World Numbers / 77 \\
6: Problems in Paradise / 95 \\
7: The Negative Numbers / 121 \\
8: Dealing with the Infinite / 135 \\
9: Dedekind's Cut: Irrational Numbers / 161 \\
10: Story of $\pi$: Transcendental Numbers / 181 \\
11: Expanding the Kingdom: Complex Numbers / 207 \\
12: Really Big: Transfinite Numbers / 223 \\
13: The Genius Calculators / 233 \\
14: What Does It All Mean? / 247 \\
15: Numbers: Past, Present, and Future / 263 \\
End Notes / 281 \\
Glossary / 289 \\
Bibliography / 299 \\
Index / 303",
}
@Article{Hauss:1994:FLC,
author = "Michael Hauss",
title = "{Fibonacci}, {Lucas}, and Central Factorial Numbers,
and $ \pi $",
journal = j-FIB-QUART,
volume = "32",
number = "5",
pages = "395--396",
month = nov,
year = "1994",
CODEN = "FIBQAU",
ISSN = "0015-0517",
ISSN-L = "0015-0517",
bibdate = "Thu Oct 20 18:02:11 MDT 2011",
bibsource = "http://www.fq.math.ca/32-5.html;
https://www.math.utah.edu/pub/tex/bib/fibquart.bib;
https://www.math.utah.edu/pub/tex/bib/pi.bib",
URL = "http://www.fq.math.ca/Scanned/32-5/hauss.pdf",
acknowledgement = ack-nhfb,
ajournal = "Fib. Quart",
fjournal = "The Fibonacci Quarterly",
journal-URL = "http://www.fq.math.ca/",
}
@TechReport{Rossner:1994:SIR,
author = "C. R{\"o}ssner and C. P. Schnorr",
title = "A stable integer relation algorithm",
type = "Report",
number = "{TR-94-016}",
institution = "FB Mathematik / Informatik Universit{\"a}t Frankfurt",
address = "Frankfurt, Germany",
pages = "1--11",
year = "1994",
bibdate = "Tue Apr 26 17:18:03 2011",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
acknowledgement = ack-nhfb,
}
@Article{Volkov:1994:CAC,
author = "Alexe{\u\i} Volkov",
title = "Calculation of $ \pi $ in ancient {China}: from {Liu
Hui} to {Zu Chongzhi}",
journal = "Historia Sci. (2)",
volume = "4",
number = "2",
pages = "139--157",
year = "1994",
ISSN = "0285-4821",
MRclass = "01A25",
MRnumber = "1325311 (96c:01014)",
MRreviewer = "Catherine Jami",
bibdate = "Mon Apr 25 16:00:23 2011",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib;
MathSciNet database",
acknowledgement = ack-nhfb,
fjournal = "Historia Scientiarum. Second Series. International
Journal of the History of Science Society of Japan",
}
@Article{Bailey:1995:FBM,
author = "David H. Bailey",
title = "A {Fortran-90} Based Multiprecision System",
journal = j-TOMS,
volume = "21",
number = "4",
pages = "379--387",
month = dec,
year = "1995",
CODEN = "ACMSCU",
DOI = "https://doi.org/10.1145/212066.212075",
ISSN = "0098-3500 (print), 1557-7295 (electronic)",
ISSN-L = "0098-3500",
bibdate = "Thu Apr 29 15:15:44 1999",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
note = "See also extension to complex arithmetic
\cite{Smith:1998:AMP}.",
URL = "http://www.acm.org/pubs/citations/journals/toms/1995-21-4/p379-bailey/",
acknowledgement = ack-rfb,
fjournal = "ACM Transactions on Mathematical Software",
journal-URL = "http://portal.acm.org/toc.cfm?idx=J782",
keywords = "arithmetic; Fortran 90; multiprecision",
ORCID-numbers = "Bailey, David H./0000-0002-7574-8342",
subject = "D.3.2 [Programming Languages]: Language
Classifications --- Fortran 90; D.3.4 [Programming
Languages]: Processors; G.1.0 [Numerical Analysis]:
General; G.1.2 [Numerical Analysis]: Approximation",
}
@Unpublished{Finch:1995:MBB,
author = "Steven Finch",
title = "The Miraculous {Bailey--Borwein--Plouffe} Pi
Algorithm",
day = "1",
month = oct,
year = "1995",
bibdate = "Tue Apr 26 15:43:06 2011",
bibsource = "https://www.math.utah.edu/pub/bibnet/authors/b/borwein-jonathan-m.bib;
https://www.math.utah.edu/pub/tex/bib/pi.bib",
note = "Recent URLs redirect to an unrelated site, but the one
given here worked on 26-Apr-2011.",
URL = "http://replay.web.archive.org/20020917121814/http://www.mathsoft.com/ASOLVE/plouffe/plouffe.html",
acknowledgement = ack-nhfb,
subject-dates = "Jonathan Michael Borwein (20 May 1951--2 August
2016)",
urlbad = "http://www.mathsoft.com/ASOLVE/plouffe/plouffe.html",
}
@Article{Hirata:1995:CTT,
author = "Keiji Hirata",
title = "Calculation of {$ \pi $} as a tool to think about the
meaning of {FGHC} programs",
journal = "S{\=u}rikaisekikenky{\=u}sho K{\=o}ky{\=u}roku",
volume = "902",
number = "??",
pages = "117--132",
month = "????",
year = "1995",
MRclass = "68N17",
MRnumber = "1372098",
bibdate = "Mon Apr 25 16:00:23 2011",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib;
MathSciNet database",
note = "The theory of parallel computation and its
applications (Japanese) (Kyoto, 1994)",
acknowledgement = ack-nhfb,
fjournal = "S{\=u}rikaisekikenky{\=u}sho K{\=o}ky{\=u}roku",
}
@Article{Rabinowitz:1995:SAD,
author = "Stanley Rabinowitz and Stan Wagon",
title = "A spigot algorithm for the digits of $ \pi $",
journal = j-AMER-MATH-MONTHLY,
volume = "102",
number = "3",
pages = "195--203",
month = mar,
year = "1995",
CODEN = "AMMYAE",
ISSN = "0002-9890 (print), 1930-0972 (electronic)",
ISSN-L = "0002-9890",
MRclass = "11Y60",
MRnumber = "96a:11152",
MRreviewer = "Andreas Guthmann",
bibdate = "Wed Dec 3 17:17:33 MST 1997",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
acknowledgement = ack-nhfb,
fjournal = "American Mathematical Monthly",
journal-URL = "https://www.jstor.org/journals/00029890.htm",
}
@Article{Adamchik:1996:PYO,
author = "Victor Adamchik and Stan Wagon",
title = "Pi: a 2000-Year-Old Search Changes Direction",
journal = "Mathematica in Science and Education",
volume = "5",
number = "1",
pages = "11--19",
month = "????",
year = "1996",
bibdate = "Sat Apr 23 09:10:07 2011",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
acknowledgement = ack-nhfb,
fjournal = "Mathematica in Science and Education",
}
@Book{Barrow:1996:PSC,
author = "John D. Barrow",
title = "Pi in the Sky: Counting, Thinking, and Being",
publisher = pub-LITTLE-BROWN,
address = pub-LITTLE-BROWN:adr,
pages = "ix + 317",
year = "1996",
ISBN = "0-316-08259-7",
ISBN-13 = "978-0-316-08259-4",
LCCN = "QA36 .B37 1994",
bibdate = "Sat Dec 17 14:44:47 MST 2005",
bibsource = "https://www.math.utah.edu/pub/tex/bib/master.bib;
https://www.math.utah.edu/pub/tex/bib/pi.bib;
z3950.loc.gov:7090/Voyager",
abstract = "John D. Barrow's \booktitle{Pi in the Sky} is a
profound -- and profoundly different exploration of the
world of mathematics: where it comes from, what it is,
and where it's going to take us if we follow it to the
limit in our search for the ultimate meaning of the
universe. Barrow begins by investigating whether math
is a purely human invention inspired by our practical
needs. Or is it something inherent in nature waiting to
be discovered? In answering these questions, Barrow
provides a bridge between the usually irreconcilable
worlds of mathematics and theology. Along the way, he
treats us to a history of counting all over the world,
from Egyptian hieroglyphics to logical friction, from
number mysticism to Marxist mathematics. And he
introduces us to a host of peculiar individuals who
have thought some of the deepest and strangest thoughts
that human minds have ever thought, from Lao-Tse to
Robert Pirsig, Charles Darwin, and Umberto Eco. Barrow
thus provides the historical framework and the
intellectual tools necessary to an understanding of
some of today's weightiest mathematical concepts.",
acknowledgement = ack-nhfb,
libnote = "Not in my library.",
remark = "Originally published: Cambridge: Oxford University,
1992.",
subject = "Mathematics",
tableofcontents = "1: From mystery to history / 1 \\
A mystery within an enigma / 1 \\
Illusions of certainty / 2 \\
The secret society / 6 \\
Non-Euclideanism / 8 \\
Logics --- To Be or Not To Be / 15 \\
The Rashomon effect / 19 \\
The analogy that never breaks down? / 21 \\
Tinkling symbols / 23 \\
Thinking about thinking / 24 \\
2: The counter culture / 26 \\
By the pricking of my thumbs / 26 \\
The bare bones of history / 28 \\
Creation or evolution / 33 \\
The ordinals versus the cardinals / 36 \\
Counting without counting / 41 \\
Fingers and toes / 45 \\
Baser methods / 49 \\
Counting with base 2 / 51 \\
The neo-2 system of counting / 56 \\
Counting In fives / 60 \\
What's so special about sixty? / 64 \\
The spread of the decimal system / 68 \\
The dance of the seven veils / 72 \\
Ritual geometry / 73 \\
The system and the Invention of zero / 81 \\
A final accounting / 101 \\
3: With form but void / 106 \\
Numerology / 106 \\
The very opposite / 108 \\
Hubert's scheme / 112 \\
Kurt G{\"o}del / 117 \\
More surprises / 124 \\
Thinking by numbers / 127 \\
Bourbachique math{\'e}matique / 129 \\
Arithmetic in chaos 1 / 34 \\
Science friction / 137 \\
Mathematics off form / 140 \\
4: The mothers of inventionism / 147 \\
Mind from matter / 147 \\
Shadowlands / 149 \\
Trap-door functions / 150 \\
Mathematical creation / 154 \\
Marxist mathematics / 156 \\
Complexity and simplicity / 159 \\
Maths as psychology / 165 \\
Pre-established mental harmony? / 171 \\
Sell-discovery / 176 \\
5: Intuitionism: the immaculate construction / 178 \\
Mathematicians from outer space / 178 \\
Ramanujan / 181 \\
Intuitionism and three-valued logic / 185 \\
A very peculiar practice / 188 \\
A closer look at Brouwer / 192 \\
What Is 'Intuition'? / 196 \\
The tragedy of Cantor and Kronecker / 198 \\
Cantor and infinity / 205 \\
The comedy of Hubert and Brouwer / 216 \\
The Four-Colour Conjecture / 227 \\
Transhuman mathematics / 234 \\
New-age mathematics / 236 \\
Paradigms / 243 \\
Computability, compressibility, and utility / 245 \\
6: Platonic heavens above and within / 249 \\
The growth of abstraction / 249 \\
Footsteps through Plato's footnotes / 251 \\
The platonic world of mathematics / 258 \\
Far away and long ago / 265 \\
The presence of the past / 268 \\
The unreasonable effectiveness of mathematics / 270 \\
Difficulties with platonic relationships / 272 \\
Seance or science? / 273 \\
Revel without a cause / 276 \\
A computer ontological argument / 280 \\
A speculative anthropic interpretation of mathematics /
284 \\
Moths and mysticism / 292 \\
Supernatural numbers? / 294 \\
further reading / 298 \\
Index / 311",
}
@TechReport{Dodge:1996:DSA,
author = "Yadolah Dodge and V. Rousson",
title = "Does $ \pi $ Satisfy all Statistical Tests?",
type = "Technical Report",
number = "96-2",
institution = "Statistics Group, University of Neuch{\^a}tel",
address = "Neuch{\^a}tel, Switzerland",
year = "1996",
bibdate = "Fri Jul 01 10:54:57 2011",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
acknowledgement = ack-nhfb,
remark = "I cannot find this report at http://www2.unine.ch/, or
in major library catalogs, or via major search
engines.",
}
@Article{Dodge:1996:NRN,
author = "Yadolah Dodge",
title = "A Natural Random Number Generator",
journal = "International Statistical Review / Revue
Internationale de Statistique",
volume = "64",
number = "3",
pages = "329--344",
month = dec,
year = "1996",
CODEN = "STRDPY",
ISSN = "0306-7734 (print), 1751-5823 (electronic)",
bibdate = "Fri Jul 01 06:59:57 2011",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
URL = "http://www.jstor.org/stable/1403789",
abstract = "Since the introduction of ``middle square'' method by
John von Neumann for the production of
``pseudo-random'' numbers in about 1949, hundreds of
other methods have been introduced. While each may have
some virtue a single uniformly superior method has not
emerged. The problems of cyclical repetition and the
need to pass statistical tests for randomness still
leave the issue unresolved. The aim of this article is
to suggest the most natural random number generator of
all, the decimals of $ \pi $, as a unique source of
random numbers. There is no cyclic behaviour, all
finite dimensional distributions of the sequence are
uniform, so that it satisfies all the properties of
today's generation of statistical tests; because of the
manner in which the numbers are generated it is
conjectured that it will satisfy any further test with
probability one. In addition, the history of $ \pi $,
its discovery and elucidation, is co-extensive with the
entire history of mankind.",
acknowledgement = ack-nhfb,
}
@Unpublished{Plouffe:1996:CTD,
author = "Simon Plouffe",
title = "On the computation of the $n$'th decimal digit of
various transcendental numbers",
day = "30",
month = nov,
year = "1996",
bibdate = "Tue Apr 26 15:48:28 2011",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
note = "The original URL no longer works, but the archive URL
retains the document.",
URL = "http://replay.web.archive.org/20021002015905/http://www.lacim.uqam.ca/plouffe/Simon/articlepi.html",
abstract = "We outline a method for computing the n'th decimal (or
any other base) digit of $ \pi $ in $ C n^3 \log (n)^3
$ time and with very little memory. The computation is
based on the recently discovered
Bailey--Borwein--Plouffe algorithm and the use of a new
algorithm that simply splits an ordinary fraction into
its components. The algorithm can be used to compute
other numbers like $ \zeta (3) $, $ \pi \sqrt {3} $, $
\pi^2 $ and $ 2 / \sqrt {5} \log (\tau) $ where $ \tau
$ is the golden ratio. The computation can be achieved
without having to compute the preceding digits. We
claim that the algorithm has a more theoretical rather
than practical interest, we have not found a faster
algorithm, nor have we proven that one does not
exist.
The formula for Pi used is $ \sum_{n = 1}^\infty n 2^n
/ {{2 n} \choose {n}} = \pi + 3 $.",
acknowledgement = ack-nhfb,
}
@Article{Wei:1996:CDD,
author = "Gong Yi Wei and Zi Giang Yang and Jia Chang Sun and
Jia Kai Li",
title = "The computation of {$ \pi $} to {$ 10, 000, 000 $}
decimal digits",
journal = j-J-NUMER-METHODS-COMPUT-APPL,
volume = "17",
number = "1",
pages = "78--81",
year = "1996",
ISSN = "1000-3266",
MRclass = "65D20",
MRnumber = "1408140",
bibdate = "Mon Apr 25 16:20:53 2011",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
note = "Also in Chinese Journal of Numerical Mathematics and
Applications, {\bf 18}(3), 96--100 (1996).",
abstract = "The algorithms of $ \pi $, the multi-precision
arithmetic operation and the fast convolution
algorithms of multi-precision multiplication are
discussed in this paper. Finally, the results of $ \pi
$ with, 8,380,000 decimal digits and 10,000,000 decimal
digits are given.",
acknowledgement = ack-nhfb,
fjournal = "Journal on Numerical Methods and Computer
Applications. Shuzhi Jisuan yu Jisuanji Yingyong",
language = "Chinese",
}
@Article{Adamchik:1997:NSF,
author = "Victor Adamchik and Stan Wagon",
title = "Notes: {A} Simple Formula for $ \pi $",
journal = j-AMER-MATH-MONTHLY,
volume = "104",
number = "9",
pages = "852--855",
month = nov,
year = "1997",
CODEN = "AMMYAE",
ISSN = "0002-9890 (print), 1930-0972 (electronic)",
ISSN-L = "0002-9890",
MRclass = "11Y60",
MRnumber = "98h:11166",
MRreviewer = "W. W. Adams",
bibdate = "Tue Jun 22 10:29:34 MDT 1999",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
note = "The authors employ Mathematica to extend earlier work
of Bailey, Borwein \cite{Borwein:1989:RME}, and
Plouffe, \cite{Bailey:1997:RCV}, done in 1995, but only
just published, that discovered an amazing formula for
$ \pi $ as is a power series in $ 16^{-k} $, enabling
any base-16 digit of $ \pi $ to be computed without
knowledge of any prior digits. In this paper,
Mathematica is used to find several simpler formulas
having powers of $ 4^{-k} $. They also note that it has
been proven that their methods cannot be used to
exhibit similar formulas in powers of $ 10^{-k} $.",
URL = "http://www.maa.org/pubs/monthly_nov97_toc.html",
acknowledgement = ack-nhfb,
fjournal = "American Mathematical Monthly",
journal-URL = "https://www.jstor.org/journals/00029890.htm",
}
@Article{Almkvist:1997:MCD,
author = "Gert Almkvist",
title = "Many correct digits of $ \pi $, revisited",
journal = j-AMER-MATH-MONTHLY,
volume = "104",
number = "4",
pages = "351--353",
month = apr,
year = "1997",
CODEN = "AMMYAE",
ISSN = "0002-9890 (print), 1930-0972 (electronic)",
ISSN-L = "0002-9890",
MRclass = "11Y60",
MRnumber = "98a:11189; 1 450 668",
MRreviewer = "Pavel Guerzhoy",
bibdate = "Tue Jun 22 10:29:34 MDT 1999",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
URL = "http://www.maa.org/pubs/monthly_apr97_toc.html",
acknowledgement = ack-nhfb,
fjournal = "American Mathematical Monthly",
journal-URL = "https://www.jstor.org/journals/00029890.htm",
}
@Article{Bailey:1997:QP,
author = "David H. Bailey and Jonathan M. Borwein and Peter B.
Borwein and Simon Plouffe",
title = "The Quest for Pi",
journal = j-MATH-INTEL,
volume = "19",
number = "1",
pages = "50--57",
month = jan,
year = "1997",
CODEN = "MAINDC",
ISSN = "0343-6993 (print), 1866-7414 (electronic)",
ISSN-L = "0343-6993",
bibdate = "Mon Apr 25 18:37:02 2011",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
acknowledgement = ack-nhfb,
author-dates = "Jonathan Michael Borwein (20 May 1951--2 August
2016)",
fjournal = "Mathematical Intelligencer",
ORCID-numbers = "Bailey, David H./0000-0002-7574-8342; Borwein,
Jonathan/0000-0002-1263-0646",
}
@Article{Bailey:1997:RCV,
author = "David Bailey and Peter Borwein and Simon Plouffe",
title = "On the rapid computation of various polylogarithmic
constants",
journal = j-MATH-COMPUT,
volume = "66",
number = "218",
pages = "903--913",
month = apr,
year = "1997",
CODEN = "MCMPAF",
ISSN = "0025-5718 (print), 1088-6842 (electronic)",
ISSN-L = "0025-5718",
MRclass = "11Yxx",
MRnumber = "1 415 794",
bibdate = "Fri Jul 16 10:38:42 MDT 1999",
bibsource = "http://www.ams.org/mcom/1997-66-218;
https://www.math.utah.edu/pub/tex/bib/pi.bib",
URL = "http://www.ams.org/jourcgi/jour-pbprocess?fn=110&arg1=S0025-5718-97-00856-9&u=/mcom/1997-66-218/",
acknowledgement = ack-nhfb,
fjournal = "Mathematics of Computation",
journal-URL = "http://www.ams.org/mcom/",
keywords = "$\pi$",
ORCID-numbers = "Bailey, David H./0000-0002-7574-8342",
}
@InCollection{Bailey:1997:RME,
author = "D. H. Bailey and J. M. Borwein and P. B. Borwein",
booktitle = "Organic mathematics ({Burnaby, BC}, 1995)",
title = "{Ramanujan}, modular equations, and approximations to
pi or {How} to compute one billion digits of pi
[{MR0991866} (90d:11143)]",
volume = "20",
publisher = pub-AMS,
address = pub-AMS:adr,
pages = "35--71",
year = "1997",
MRclass = "11Y60 (01A60 11F03 33E20)",
MRnumber = "1483913",
bibdate = "Fri Jan 9 13:11:19 2015",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
series = "CMS Conference Proceedings",
acknowledgement = ack-nhfb,
author-dates = "Jonathan Michael Borwein (20 May 1951--2 August
2016)",
ORCID-numbers = "Bailey, David H./0000-0002-7574-8342; Borwein,
Jonathan/0000-0002-1263-0646",
}
@InProceedings{Bailey:1997:RNC,
author = "David H. Bailey and Simon Plouffe",
booktitle = "The Organic Mathematics Project Proceedings",
title = "Recognizing Numerical Constants",
volume = "20",
publisher = "Canadian Mathematical Society",
address = "Ottawa, ON K1G 3V4, Canada",
pages = "73--88",
month = "????",
year = "1997",
CODEN = "CJMAAB",
ISSN = "0008-414X (print), 1496-4279 (electronic)",
bibdate = "Tue Apr 26 15:57:14 2011",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
URL = "http://crd.lbl.gov/~dhbailey/dhbpapers/recog.pdf;
http://www.cecm.sfu.ca/organics",
abstract = "The advent of inexpensive, high-performance computers
and new efficient algorithms have made possible the
automatic recognition of numerically computed
constants. In other words, techniques now exist for
determining, within certain limits, whether a computed
real or complex number can be written as a simple
expression involving the classical constants of
mathematics.\par
These techniques will be illustrated by discussing the
recognition of Euler sum constants, and also the
discovery of new formulas for $ \pi $ and other
constants, formulas that permit individual digits to be
extracted from their expansions.",
acknowledgement = ack-nhfb,
keywords = "PSLQ algorithm",
ORCID-numbers = "Bailey, David H./0000-0002-7574-8342",
}
@Unpublished{Bellard:1997:BBD,
author = "Fabrice Bellard",
title = "The 1000 billionth binary digit of pi is `1'!",
year = "1997",
bibdate = "Tue Apr 26 09:36:33 2011",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
note = "Was this work published elsewhere?",
URL = "http://bellard.org/pi-challenge/announce220997.html",
acknowledgement = ack-nhfb,
remark = "Calculation took 12 days on 20 workstations, and 180
CPU days.",
}
@Unpublished{Bellard:1997:NFC,
author = "Fabrice Bellard",
title = "A new formula to compute the $n$-th binary digit of $
\pi $",
year = "1997",
bibdate = "Tue Apr 26 09:36:33 2011",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
note = "This formula is claimed in \cite{Sze:2010:TQB} to be
somewhat faster to compute than the BBP formula.",
URL = "http://bellard.org/pi/pi_bin.pdf",
acknowledgement = ack-nhfb,
}
@Book{Blatner:1997:JP,
author = "David Blatner",
title = "The Joy of $ \pi $",
publisher = "Walker and Co.",
address = "New York, NY, USA",
pages = "xiii + 129",
year = "1997",
ISBN = "0-8027-1332-7 (hardcover), 0-8027-7562-4 (paperback)",
ISBN-13 = "978-0-8027-1332-2 (hardcover), 978-0-8027-7562-7
(paperback)",
LCCN = "QA484 .B55 1997",
bibdate = "Fri Jun 17 06:26:55 MDT 2005",
bibsource = "https://www.math.utah.edu/pub/tex/bib/master.bib;
https://www.math.utah.edu/pub/tex/bib/pi.bib;
z3950.loc.gov:7090/Voyager",
URL = "http://www.walkerbooks.com/books/catalog.php?key=4",
abstract = "No number has captured the attention and imagination
of people throughout the ages as much as the ratio of a
circle's circumference to its diameter. Pi or $ \pi $
as it is symbolically known, is infinite and, in this
book it proves to be infinitely intriguing. The author
explores the many facets of pi and humankind's
fascination with it, from the ancient Egyptians and
Archimedes to Leonardo da Vinci and the modern-day
Chudnovsky brothers, who have calculated pi to eight
billion digits with a homemade supercomputer. He
recounts the history of pi and the quirky stories of
those obsessed with it. Sidebars document fascinating
pi trivia (including a segment from the O. J. Simpson
trial). Dozens of snippets and factoids reveal pi's
remarkable impact over the centuries. Mnemonic devices
teach how to memorize pi to many hundreds of digits (or
more, if you're so inclined). Pi inspired cartoons,
poems, limericks, and jokes offer delightfully
``square'' pi humor. And, to satisfy even the most
exacting of number jocks, the first one million digits
of pi appear throughout the book.",
acknowledgement = ack-nhfb,
libnote = "Not in my library.",
subject = "Pi (mathematical constant)",
tableofcontents = "1: Introduction: Why pi \\
2: History of pi \\
3: Chudnovsky brothers \\
4: Symbol \\
5: Personality of pi \\
6: Circle squarers \\
7: Memorizing pi",
}
@Book{Delahaye:1997:FNc,
author = "Jean-Paul Delahaye",
title = "Le fascinant nombre $ \pi $ ({French}) [{The}
fascinating number $ \pi $]",
publisher = "{\'E}ditions Belin / Pour La Science",
address = "Paris, France",
pages = "224",
year = "1997",
ISBN = "2-902918-25-9",
ISBN-13 = "978-2-902918-25-6",
ISSN = "0224-5159",
ISSN-L = "0224-5159",
LCCN = "QA484 D44 1997",
bibdate = "Mon Jun 15 07:45:06 MDT 2020",
bibsource = "fsz3950.oclc.org:210/WorldCat;
https://www.math.utah.edu/pub/tex/bib/pi.bib",
series = "Biblioth{\'y}eque Pour la science",
abstract = "Le nombre pi est au centre d'un cercle
math{\'e}matique extraordinaire. Cette {\'e}tude
retrace l'histoire de son exploration en insistant sur
les {\'e}pisodes les plus r{\'e}cents. Apr{\'e}s 4000
ans de travail et de d{\'e}couverte, les
math{\'e}maticiens arrivent encore {\`a} trouver de
nouvelles propri{\'e}t{\'e}s de pi.",
acknowledgement = ack-nhfb,
language = "French",
subject = "Pi (le nombre); Math{\'e}matiques; Histoire;
Geschichte; Zahl; Pi (Le nombre); Histoire; Nombre Pi;
origines",
}
@Article{Laczkovich:1997:LPI,
author = "M. Laczkovich",
title = "On {Lambert}'s proof of the irrationality of $ \pi $",
journal = j-AMER-MATH-MONTHLY,
volume = "104",
number = "5",
pages = "439--443",
month = may,
year = "1997",
CODEN = "AMMYAE",
ISSN = "0002-9890 (print), 1930-0972 (electronic)",
ISSN-L = "0002-9890",
MRclass = "11J72 (11A55)",
MRnumber = "98a:11090; 1 447 977",
MRreviewer = "Carsten Elsner",
bibdate = "Tue Jun 22 10:29:34 MDT 1999",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
note = "See \cite{Lambert:1768:MQP}.",
URL = "http://www.jstor.org/stable/2974737;
http://www.maa.org/pubs/monthly_may97_toc.html",
acknowledgement = ack-nhfb,
fjournal = "American Mathematical Monthly",
journal-URL = "https://www.jstor.org/journals/00029890.htm",
}
@Article{Ogawa:1997:BEC,
author = "Tsukane Ogawa",
title = "The beginnings of enri---the calculation of $ \pi $ by
{Katahiro Takebe}",
journal = "S{\=u}rikaisekikenky{\=u}sho K{\=o}ky{\=u}roku",
volume = "1019",
number = "??",
pages = "77--97",
year = "1997",
MRclass = "01A45",
MRnumber = "1648905",
bibdate = "Mon Apr 25 16:00:23 2011",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib;
MathSciNet database",
note = "Study of the history of mathematics (Japanese) (Kyoto,
1997)",
acknowledgement = ack-nhfb,
fjournal = "S{\=u}rikaisekikenky{\=u}sho K{\=o}ky{\=u}roku",
}
@Article{Volkov:1997:ZYH,
author = "Alexe{\"u\i} Volkov",
title = "{Zhao Youqin} and his calculation of $ \pi $",
journal = j-HIST-MATH,
volume = "24",
number = "3",
pages = "301--331",
month = aug,
year = "1997",
CODEN = "HIMADS",
DOI = "https://doi.org/10.1006/hmat.1997.2163",
ISSN = "0315-0860 (print), 1090-249X (electronic)",
ISSN-L = "0315-0860",
MRclass = "01A25",
MRnumber = "1470103 (98g:01015)",
bibdate = "Wed Jun 26 06:19:20 MDT 2013",
bibsource = "https://www.math.utah.edu/pub/tex/bib/histmath.bib;
https://www.math.utah.edu/pub/tex/bib/pi.bib; MathSciNet
database",
URL = "http://www.sciencedirect.com/science/article/pii/S0315086097921637",
abstract = "The paper discusses the method used by Zhao Youqin
(1271--?) in his treatise ``Ge xiang xin shuto'' to
confirm Zu Chongzhi's (429--500) approximate value $
355 / 113 $ of $ \pi $. Zhao Youqin inscribed a square
into a circle and performed an iterative procedure of
calculation of one side of a $ 2 n $-sided inscribed
polygon for $ n = 3, \ldots {}, 14 $. Included is a
biographical sketch of Zhao Youqin, who was an
astronomer, mathematician, and physicist as well as a
Taoist monk and alchemist. A translation of Zhao's
description of his method is given in the Appendix.",
acknowledgement = ack-nhfb,
fjournal = "Historia Mathematica",
journal-URL = "http://www.sciencedirect.com/science/journal/03150860",
}
@Book{Arndt:1998:ACA,
author = "J{\"o}rg Arndt and Christoph Haenel",
title = "{$\pi$: Algorithmen, Computer, Arithmetik}",
publisher = pub-SV,
address = pub-SV:adr,
pages = "xi + 191",
year = "1998",
ISBN = "3-540-63419-3",
ISBN-13 = "978-3-540-63419-5",
LCCN = "????",
bibdate = "Mon Jun 15 07:03:04 MDT 2020",
bibsource = "fsz3950.oclc.org:210/WorldCat;
https://www.math.utah.edu/pub/tex/bib/pi.bib",
URL = "http://zbmath.org/?q=an:0893.11001",
acknowledgement = ack-nhfb,
language = "German",
tableofcontents = "1. Der Stand der Dinge / 1 \\
2. Wie zuf{\"a}llig ist $\pi$? / 13 \\
2.1 Wahrscheinlichkeiten / 13 \\
2.2 Ist $\pi$ normal? / 13 \\
2.3 Doch nicht normal? / 16 \\
2.4 Weitere statistische Ergebnisse / 18 \\
3. Leichte Wege zu $\pi$ / 23 \\
3.1 Kannitverstahn / 23 \\
3.2 Monte Carlo-Verfahren / 25 \\
3.3 Memorabilia / 29 \\
3.4 Die fr{\"u}heste Kreisquadratur der Geschichte? /
31 \\
3.5 Verbesserungen / 33 \\
3.6 Der $\pi$-Saal in Paris / 34 \\
4. N{\"a}herungen f{\"u}r $\pi$ und Kettenbr{\"u}che /
35 \\
4.1 N{\"a}herungen / 35 \\
4.2 {\"U}ber Kettenbr{\"u}che / 43 \\
5. Arcus Tangens / 47 \\
5.1 Die arctan-Formel von John Machin / 47 \\
5.2 Weitere arctan-Formeln / 50 \\
6. Tr{\"o}pfel-Algorithmen / 55 \\
6.1 Ein Mini-C-Programm f{\"u}r $\pi$ / 56 \\
6.2 Der Tr{\"o}pfel-Algorithmus im Detail / 56 \\
6.3 Ablauf / 58 \\
6.4 Eine einfachere Variante / 59 \\
6.5 Tr{\"o}pfel-Algorithmus f{\"u}r $e$ / 61 \\
7. Gau{\ss} und $\pi$ / 63 \\
7. 1 Die $\pi$-AGM-Formel / 63 \\
7.2 Der Gau{\ss}-AGM-Algorithmus / 66 \\
7.3 Historie einer Formel / 68 \\
8. Ramanujan und $\pi$ / 75 \\
8.1 Ramanujansche Reihen / 75 \\
8.2 Ramanujans ungew{\"o}hnliche Biographie / 77 \\
8.3 Impulse / 81 \\
9. Die Borweins und $\pi$ / 83 \\
10. Das BBP-Verfahren / 87 \\
10.1 Bin{\"a}re Modulo-Exponentation / 90 \\
10.2 Ein C-Programm zur BBP-Reihe / 92 \\
10.3 Verbesserungen / 95 \\
11. Arithmetik / 99 \\
11.1 Karatsuba Multiplikation / 100 \\
11.2 Schnelle Fourier-Multiplikation / 101 \\
11.3 Division / 104 \\
11.4 Berechnung von Quadratwurzeln / 105 \\
12. Vermischtes / 107 \\
12.1 Ein Pi-Quiz / 107 \\
12.2 La{\ss}t Zahlen sprechen / 108 \\
12.3 Ein Beweis f{\"u}r $\pi = 2$ / 109 \\
12.4 The Big Change/ 109 \\
12.5 Fast voll daneben / 109 \\
12.6 Warum immer mehr Stellen? / 111 \\
12.7 Kreisquadratur mit L{\"o}chern / 111 \\
13. Historie / 115 \\
13.1 Altertum / 116 \\
13.2 Archimedes und die zwei Jahrtausende danach / 117
\\
13.3 Unendliche Reihen / 123 \\
13.4 Hochleistungsalgorithmen / 131 \\
13.5 Ein $\pi$-Gesetz / 137 \\
13.6 Der Fall Bieberbach / 138 \\
13.7 Ein fr{\"u}her (Fast-) Weltrekord / 139 \\
14. Die Zukunft: Internet $\pi$-Berechnungen / 143 \\
14.1 Der binary splitting (binsplit) Algorithmus / 143
\\
14.2 Das Internet $\pi$-Projekt / 146 \\
15. Formelsammlung $\pi$ / 149 \\
16. Tabellen / 163 \\
16.1 Ausgew{\"a}hlte Konstante auf 100 Stellen (Basis
10) / 163 \\
16.2 Die ersten 2.500 Stellen von $\pi$ (Basis 10) /
164 \\
16.3 Ausgew{\"a}hlte Konstante auf 100 Stellen (Basis
16) / 165 \\
16.4 Die ersten 2.500 Stellen von $\pi$ (Basis 16) /
166 \\
A. Documentation for the hfloat library / 167 \\
A.1 What hfloat is (good for) / 167 \\
A.2 Compiling the library / 168 \\
A.3 Functions of the hfloat library / 168 \\
A.4 Using hfloats in your own code / 170 \\
A.5 Computations with extreme precision / 173 \\
A.6 Precision and radix / 173 \\
A.7 Compiling \& running the $\pi$-example code / 174
\\
A.8 Structure of hfloat / 175 \\
A.9 Organisation of the files / 176 \\
A.10 Distribution policy \& no warranty / 177 \\
B. Other high precision libraries / 179 \\
Literaturverzeichnis / 181 \\
Index / 187",
}
@Article{Bailey:1998:FNM,
author = "David H. Bailey",
title = "Finding New Mathematical Identities via Numerical
Computations",
journal = j-SIGNUM,
volume = "33",
number = "1",
pages = "17--22",
month = jan,
year = "1998",
CODEN = "SNEWD6",
DOI = "https://doi.org/10.1145/381866.381887",
ISSN = "0163-5778 (print), 1558-0237 (electronic)",
ISSN-L = "0163-5778",
bibdate = "Tue Apr 12 07:50:30 MDT 2005",
bibsource = "http://portal.acm.org/;
https://www.math.utah.edu/pub/tex/bib/pi.bib",
abstract = "A recent development in computational mathematics is
the use of high-precision numerical computations,
together with advanced integer relation algorithms, to
discover heretofore unknown mathematical identities.
One of these new identities, a remarkable new formula
for $ \pi $, permits one to directly compute the $n$-th
hexadecimal digit of $ \pi $, without computing the
first $ n - 1$ digits, and without the need of
multiple-precision arithmetic software.",
acknowledgement = ack-nhfb,
fjournal = "ACM SIGNUM Newsletter",
keywords = "BBP (Bailey, Borwein, Plouffe) formula; PSLQ
algorithm",
ORCID-numbers = "Bailey, David H./0000-0002-7574-8342",
}
@Unpublished{Borwein:1998:TAP,
author = "Jonathan Borwein",
title = "Talking about Pi",
day = "20",
month = jan,
year = "1998",
bibdate = "Tue Apr 26 18:14:36 2011",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
note = "The original URL is no longer found, but the archive
URL worked on 26-Apr-2011.",
acknowledgement = ack-nhfb,
author-dates = "Jonathan Michael Borwein (20 May 1951--2 August
2016)",
ORCID-numbers = "Borwein, Jonathan/0000-0002-1263-0646",
}
@Article{Smith:1998:AMP,
author = "David M. Smith",
title = "{Algorithm 786}: Multiple-Precision Complex Arithmetic
and Functions",
journal = j-TOMS,
volume = "24",
number = "4",
pages = "359--367",
month = dec,
year = "1998",
CODEN = "ACMSCU",
DOI = "https://doi.org/10.1145/293686.293687",
ISSN = "0098-3500 (print), 1557-7295 (electronic)",
ISSN-L = "0098-3500",
bibdate = "Tue Mar 09 10:09:51 1999",
bibsource = "http://www.acm.org/pubs/contents/journals/toms/1998-24/;
https://www.math.utah.edu/pub/tex/bib/pi.bib",
note = "See also
\cite{Bailey:1995:FBM,Brent:1978:AMF,Brent:1979:RMF,Brent:1980:AIB}.",
URL = "http://www.acm.org:80/pubs/citations/journals/toms/1998-24-4/p359-smith/",
abstract = "The article describes a collection of Fortran routines
for multiple-precision complex arithmetic and
elementary functions. The package provides good
exception handling, flexible input and output, trace
features, and results that are almost always correctly
rounded. For best efficiency on different machines, the
user can change the arithmetic type used to represent
the multiple-precision numbers.",
acknowledgement = ack-nhfb,
fjournal = "ACM Transactions on Mathematical Software",
journal-URL = "http://portal.acm.org/toc.cfm?idx=J782",
keywords = "algorithms; performance; reliability",
subject = "{\bf G.1.0} Mathematics of Computing, NUMERICAL
ANALYSIS, General, Computer arithmetic. {\bf G.1.2}
Mathematics of Computing, NUMERICAL ANALYSIS,
Approximation, Elementary function approximation. {\bf
G.4} Mathematics of Computing, MATHEMATICAL SOFTWARE,
Algorithm design and analysis. {\bf G.4} Mathematics of
Computing, MATHEMATICAL SOFTWARE, Efficiency. {\bf G.4}
Mathematics of Computing, MATHEMATICAL SOFTWARE,
Portability**.",
}
@InCollection{Symborska:1998:P,
author = "Wis{\l}awa Symborska",
booktitle = "Poems, New and Collected, 1957--1997",
title = "{PI}",
publisher = "Harcourt Brace",
address = "New York, NY, USA",
bookpages = "xvii + 273",
pages = "174--175",
year = "1998",
ISBN = "0-15-100353-X",
ISBN-13 = "978-0-15-100353-2",
LCCN = "PG7178.Z9 A222 1998",
bibdate = "Mon Jun 10 08:31:41 2013",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
note = "Translated from the Polish by Stanis{\l}aw
Bara{\'n}czak and Clare Cavanagh.",
URL = "http://www.nobelprize.org/nobel_prizes/literature/laureates/1996/;
http://www.nobelprize.org/nobel_prizes/literature/laureates/1996/szymborska.html",
acknowledgement = ack-nhfb,
authordates = "2 July 1923--1 February 2012",
remark = "The author is the winner of the 1996 Nobel Prize in
Literature ``for poetry that with ironic precision
allows the historical and biological context to come to
light in fragments of human reality.''",
}
@Article{Takahashi:1998:CBD,
author = "Daisuke Takahashi and Yasumasa Kanada",
title = "Calculation of $ \pi $ to 51.5 billion decimal digits
on distributed memory parallel processors",
journal = j-TRANS-INFO-PROCESSING-SOC-JAPAN,
volume = "39",
number = "7",
pages = "2074--2083",
year = "1998",
CODEN = "JSGRD5",
ISSN = "0387-5806",
ISSN-L = "0387-5806",
MRclass = "65D20 (11Y60)",
MRnumber = "1639333 (99d:65063)",
bibdate = "Mon Apr 25 16:00:23 2011",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib;
MathSciNet database",
acknowledgement = ack-nhfb,
fjournal = "Information Processing Society of Japan.
Transactions",
}
@Article{Tsaban:1998:RAP,
author = "Boaz Tsaban and David Garber",
title = "On the {Rabbinical} Approximation of $ \pi $",
journal = j-HIST-MATH,
volume = "25",
number = "1",
pages = "75--84",
month = feb,
year = "1998",
CODEN = "HIMADS",
ISSN = "0315-0860 (print), 1090-249X (electronic)",
ISSN-L = "0315-0860",
bibdate = "Wed Jun 26 06:19:26 MDT 2013",
bibsource = "https://www.math.utah.edu/pub/tex/bib/histmath.bib;
https://www.math.utah.edu/pub/tex/bib/pi.bib",
URL = "http://www.sciencedirect.com/science/article/pii/S0315086097921856",
acknowledgement = ack-nhfb,
fjournal = "Historia Mathematica",
journal-URL = "http://www.sciencedirect.com/science/journal/03150860",
}
@Article{Ferguson:1999:API,
author = "Helaman R. P. Ferguson and David H. Bailey and Steve
Arno",
title = "Analysis of {PSLQ}, an integer relation finding
algorithm",
journal = j-MATH-COMPUT,
volume = "68",
number = "225",
pages = "351--369",
month = jan,
year = "1999",
CODEN = "MCMPAF",
ISSN = "0025-5718 (print), 1088-6842 (electronic)",
ISSN-L = "0025-5718",
MRclass = "11Y16 (68Q25)",
MRnumber = "1 489 971",
bibdate = "Fri Jul 16 10:39:00 MDT 1999",
bibsource = "http://www.ams.org/mcom/1999-68-225;
https://www.math.utah.edu/pub/tex/bib/pi.bib",
URL = "http://www.ams.org/jourcgi/jour-pbprocess?fn=110&arg1=S0025-5718-99-00995-3&u=/mcom/1999-68-225/",
abstract = "Let $ {\mathbb {K}} $ be either the real, complex, or
quaternion number system and let $ {\mathbb
{O}}({\mathbb {K}}) $ be the corresponding integers.
Let $ x = (x_1, \ldots, x_n) $ be a vector in $
{\mathbb {K}}^n $. The vector $x$ has an integer
relation if there exists a vector $ m = (m_1, \ldots,
m_n) \in {\mathbb {O}}({\mathbb {K}})^n$, $ m \ne 0$,
such that $ m_1 x_1 + m_2 x_2 + \ldots + m_n x_n = 0$.
In this paper we define the parameterized integer
relation construction algorithm PSLQ$ (\tau)$, where
the parameter $ \tau $ can be freely chosen in a
certain interval. Beginning with an arbitrary vector $
x = (x_1, \ldots, x_n) \in {\mathbb {K}}^n$, iterations
of PSLQ$ (\tau)$ will produce lower bounds on the norm
of any possible relation for $x$. Thus PSLQ$ (\tau)$
can be used to prove that there are no relations for
$x$ of norm less than a given size. Let $ M_x$ be the
smallest norm of any relation for $x$. For the real and
complex case and each fixed parameter $ \tau $ in a
certain interval, we prove that PSLQ$ (\tau)$
constructs a relation in less than $ O(n^3 + n^2 \log
M_x)$ iterations.",
acknowledgement = ack-nhfb,
fjournal = "Mathematics of Computation",
journal-URL = "http://www.ams.org/mcom/",
ORCID-numbers = "Bailey, David H./0000-0002-7574-8342",
}
@Misc{Gourdon:1999:NCC,
author = "Xavier Gourdon and Pascal Sebah",
title = "Numbers, constands, and computation",
howpublished = "Web site",
year = "1999",
bibdate = "Mon Jun 15 07:52:53 2020",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
URL = "http://numbers.computation.free.fr/Constants/constants.html",
acknowledgement = ack-nhfb,
}
@Misc{Gourdon:1999:PEU,
author = "X. Gourdon",
title = "{PiFast}, an easy-to-use package for computing pi and
other irrationals to large numbers of digits",
howpublished = "Web site.",
year = "1999",
bibdate = "Fri Jul 01 06:43:52 2011",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
URL = "http://www.numbers.computation.free.fr/Constants/PiProgram/pifast.html",
acknowledgement = ack-nhfb,
}
@Unpublished{Group:1999:P,
author = "{Pi Group}",
title = "The {$ \pi $} Pages",
day = "8",
month = may,
year = "1999",
bibdate = "Tue Apr 26 18:11:25 2011",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
note = "The original URL is no longer found, but the archive
URL worked on 26-Apr-2011.",
URL = "http://replay.web.archive.org/20020812145823/http://www.cecm.sfu.ca/PI/",
acknowledgement = ack-nhfb,
}
@Article{Lange:1999:NEC,
author = "L. J. Lange",
title = "Notes: An Elegant Continued Fraction for $ \pi $",
journal = j-AMER-MATH-MONTHLY,
volume = "106",
number = "5",
pages = "456--458",
month = may,
year = "1999",
CODEN = "AMMYAE",
DOI = "https://doi.org/10.2307/2589152",
ISSN = "0002-9890 (print), 1930-0972 (electronic)",
ISSN-L = "0002-9890",
bibdate = "Sat Sep 11 08:13:57 1999",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
acknowledgement = ack-nhfb,
fjournal = "American Mathematical Monthly",
journal-URL = "https://www.jstor.org/journals/00029890.htm",
}
@Article{Lord:1999:RFA,
author = "Nick Lord",
title = "83.50 Recent Formulae for $ \pi $: Arctan Revisited!",
journal = j-MATH-GAZ,
volume = "83",
number = "498",
publisher = "JSTOR",
pages = "479--483",
year = "1999",
CODEN = "MAGAAS",
ISSN = "0025-5572 (print), 2056-6328 (electronic)",
bibdate = "Thu Jun 9 15:45:09 2016",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
acknowledgement = ack-nhfb,
fjournal = "Mathematical Gazette",
journal-URL = "http://www.m-a.org.uk/jsp/index.jsp?lnk=620",
}
@Book{Arndt:2000:ACA,
author = "J{\"o}rg Arndt and Christoph Haenel",
title = "{$ \pi $: Algorithmen, Computer, Arithmetik}",
publisher = pub-SV,
address = pub-SV:adr,
edition = "Second",
year = "2000",
DOI = "https://doi.org/10.1007/978-3-662-09360-3",
ISBN = "3-540-66258-8 (print), 3-662-09360-X",
ISBN-13 = "978-3-540-66258-7 (print), 978-3-662-09360-3",
LCCN = "QA76.9.A43",
bibdate = "Mon Jun 15 07:37:13 MDT 2020",
bibsource = "fsz3950.oclc.org:210/WorldCat;
https://www.math.utah.edu/pub/tex/bib/pi.bib",
URL = "http://link.springer.com/openurl?genre=book\%26isbn=978-3-540-66258-7",
abstract = "Ausgehend von der Programmierung moderner
Hochleistungsalgorithmen stellen die Autoren das
mathematische und programmtechnische Umfeld der Zahl Pi
ausf{\"u}hrlich dar. So werden zur Berechnung von Pi
sowohl die arithmetischen Algorithmen, etwa die
FFT-Multiplikation, die super-linear konvergenten
Verfahren von Gau{\ss}, Brent, Salamin, Borwein, die
Formeln von Ramanujan und Borwein--Bailey--Plouffe bis
zum neuen Tr{\"o}pfel-Algorithmus behandelt. Der Leser
findet viel Anregendes wie auch Skurriles, etwa
interessante Anmerkungen zur Quadratur des Kreises. Die
beigelegte CD-ROM bietet dem User mannigfaltigen
Nutzen, z. B. die ausgef{\"u}hrte Langzahlarithmetik
hfloat im C++ Source-Code, die FFT-Multiplikation und
Algorithmen zur Pi-Berechnung. Die zweite,
{\"u}berarbeitete Auflage nimmt zahlreiche
Leseranregungen auf und berichtet {\"u}ber die
wichtigsten neuesten Ergebnisse der Pi-Forschung.
Zahlreiche Verweise auf Internetquellen,
ausf{\"u}hrlicher Index und Literaturverzeichnis
erg{\"a}nzen das Buch.",
acknowledgement = ack-nhfb,
language = "German",
subject = "Computer science; Computer software; Algebra; Data
processing; Number theory",
tableofcontents = "1. Der Stand der Dinge \\
2. Wie zuf{\"a}llig ist? \\
3. Leichte Wege zu? \\
4. N{\"a}herungen f{\"u}r und Kettenbr{\"u}che \\
5. Arcus Tangens \\
6. Tr{\"o}pfel-Algorithmen \\
7. Gau{\ss} und? \\
8. Ramanujan und? \\
9. Die Borweins und? \\
10. Das BBP-Verfahren \\
11. Arithmetik \\
12. Vermischtes \\
13. Die Historie von? \\
14. Historische Notizen \\
15. Die Zukunft: --- Berechnungen im Internet \\
16. Formelsammlung \\
17. Tabellen \\
A. Documentation for the hfloat-library \\
A.1 What hfloat is (good for) \\
A.2 Compiling the library \\
A.3 Functions of the hfloat-library \\
A.4 Using hfloats in your own code \\
A.5 Computations with extreme precision \\
A.6 Precision and radix \\
A.7 Compiling and running the example code \\
A.8 Structure of hfloat \\
A.9 Organisation of the files \\
A.10 Distribution policy and no warranty \\
B. Other high precision libraries",
}
@Article{Bailey:2000:IRD,
author = "David H. Bailey",
title = "Integer Relation Detection",
journal = j-COMPUT-SCI-ENG,
volume = "2",
number = "1",
pages = "24--28",
month = jan # "\slash " # feb,
year = "2000",
CODEN = "CSENFA",
DOI = "https://doi.org/10.1109/5992.814653",
ISSN = "1521-9615 (print), 1558-366X (electronic)",
ISSN-L = "1521-9615",
bibdate = "Fri Oct 13 14:31:09 2000",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
URL = "http://dlib.computer.org/cs/books/cs2000/pdf/c1024.pdf;
http://www.computer.org/cse/cs1999/c1024abs.htm",
abstract = "Practical algorithms for integer relation detection
have become a staple in the emerging discipline of
``experimental mathematics'' --- using modern computer
technology to explore mathematical research. After
briefly discussing the problem of integer relation
detection, the author describes several recent,
remarkable applications of these techniques in both
mathematics and physics.",
acknowledgement = ack-nhfb,
fjournal = "Computing in Science and Engineering",
journal-URL = "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=5992",
keywords = "PSLQ algorithm",
ORCID-numbers = "Bailey, David H./0000-0002-7574-8342",
}
@Book{Hardy:2000:CPS,
editor = "G. H. (Godfrey Harold) Hardy and P. V. (P.
Venkatesvara) {Seshu Aiyar} and B. M. (Bertram Martin)
Wilson",
title = "Collected papers of {Srinivasa Ramanujan}",
publisher = "AMS Chelsea Publishing Company",
address = "Providence, RI, USA",
pages = "xxxviii + 426",
year = "2000",
ISBN = "0-8218-2076-1 (hardcover)",
ISBN-13 = "978-0-8218-2076-6 (hardcover)",
LCCN = "QA3 .S685 1962",
MRclass = "11-06, 01A75, 33-06, 084 $5 335222209 $a 01A75",
bibdate = "Fri Jan 9 12:57:23 MST 2015",
bibsource = "fsz3950.oclc.org:210/WorldCat;
https://www.math.utah.edu/pub/tex/bib/pi.bib",
acknowledgement = ack-nhfb,
remark = "Originally published as \cite{Hardy:1927:CPS}.
Reprinted as \cite{Hardy:1962:CPS}. With new preface
and commentary.",
subject = "Zahlentheorie; Geschichte 1911--1921;
Aufsatzsammlung",
tableofcontents = "Preface \\
Notice / P. V. Seshu Aiyar and R. Bamachaundra Rao \\
Notice / G. H. Hardy \\
Preface to the Third Printing / Bruce C. Berndt \\
Papers \\
1: Some properties of Bernoulli's numbers [J. Indian
Math. Soc. 3 (1911), 219--234] / 1--14\\
2: On Question 330 of Prof. Sanjana [J. Indian Math.
Soc. 4 (1912), 59--61] / 15--17 \\
3: Note on a set of simultaneous equations [J. Indian
Math. Soc. 4 (1912), 94--96] / 18--19 \\
4: Irregular numbers [J. Indian Math. Soc. 5 (1913),
105--106] / 20--21 \\
5: Squaring the circle [J. Indian Math. Soc. 5 (1913),
132] / 22 \\
6: Modular equations and approximations to $\pi$
[Quart. J. Math. 45 (1914), 350--372] / 23--39 \\
7: On the integral $\int_0^x
\frac{\tan^{-1}(t)}{t}\,dt$ [J. Indian Math. Soc. 7
(1915), 93--96] / 40--43 \\
8: On the number of divisors of a number [J. Indian
Math. Soc. 7 (1915), 131--133] / 44--46 \\
9: On the sum of the square roots of the first $n$
natural numbers [J. Indian Math. Soc. 7 (1915),
173--175] / 47--49 \\
10: On the product $\Prod_{n = 0}^{n = \infty} \left[ 1
+ (\left(\frac{x}{a + n d}\right)^3 \right]$ [J. Indian
Math. Soc. 7 (1915), 209--211] / 50--52 \\
11: Some definite integrals [Messenger Math. 44 (1915),
10--18] / 53--58 \\
12: Some definite integrals connected with Gauss's sums
[Messenger Math. 44 (1915), 75--85] / 59--67 \\
13: Summation of a certain series [Messenger Math. 44
(1915), 157--160] / 68--71 \\
14: New expressions for Riemann's functions $\xi(s)$
and $\Xi(t)$ [Quart. J. Math. 46 (1915), 253--260] /
72--77 \\
15: Highly composite numbers [Proc. London Math. Soc.
(2) 14 (1915), 347--409] / 78--128 \\
16: On certain infinite series [Messenger Math. 45
(1916) 11--15] / 129--132 \\
17: Some formul{\ae} in the analytic theory of numbers
[Messenger Math. 45 (1916), 81--84] / 133--135 \\
18: On certain arithmetical functions [Trans. Cambridge
Philos. Soc. 22 (1916), no. 9, 159--184] / 136--162 \\
19: A series for Euler's constant $\gamma$ [Messenger
Math. 46 (1917), 73--80] / 163--168 \\
20: On the expression of a number in the form
$ax^2+by^2+cz^2+du^2$ [Proc. Cambridge Philos. Soc. 19
(1917), 11--21] / 169--178 \\
21: On certain trigonometrical sums and their
applications in the theory of numbers [Trans. Cambridge
Philos. Soc. 22 (1918), no. 13, 259--276] / 179--199
\\
22: Some definite integrals [Proc. London Math. Soc.
(2) 17 (1918), Records for 17 Jan. 1918] / 200--201 \\
23: Some definite integrals [J. Indian Math. Soc. 11
(1919), 81--87] / 202--207 \\
24: A proof of Bertrand's postulate [J. Indian Math.
Soc. 11 (1919), 181--182] / 208--209 \\
25: Some properties of $p(n)$, the number of partitions
of $n$ [Proc. Cambridge Philos. Soc. 19 (1919),
207--210] / 210--213 \\
26: Proof of certain identities in combinatory analysis
[Proc. Cambridge Philos. Soc. 19 (1919), 214--216] /
214--215 \\
27: A class of definite integrals [Quart. J. Math. 48
(1920), 294--310] / 216--229 \\
28: Congruence properties of partitions [Proc. London
Math. Soc. (2) 18 (1920), Records for 13 March 1919] /
230 \\
\\
29: Algebraic relations between certain infinite
products [Proc. London Math. Soc. (2) 18 (1920),
Records for 13 March 1919] / 231 \\
30: Congruence properties of partitions / 232--238 \\
31: G. H. Hardy and S. Ramanujan / Une formule
asymptotique pour le nombre des partitions de $n$
[Comptes Rendus, 2 Jan. 1917] [An asymptotic formula
for the number of partitions of $n$] / 239--241 \\
32: G. H. Hardy and S. Ramanujan / Proof that almost
all numbers $n$ are composed of about $\log\log n$
prime factors [Proc. London Math. Soc. (2) 16 (1917),
Records for 14 Dec. 1916] / 242--243 \\
33: G. H. Hardy and S. Ramanujan / Asymptotic
formul{\ae} in combinatory analysis [Proc. London Math.
Soc. (2) 16 (1917), Records for 1 March 1917] / 244 \\
34: G. H. Hardy and S. Ramanujan / Asymptotic
formul{\ae} for the distribution of integers of various
types [Proc. London Math. Soc. (2) 16 (1917), 112--132]
/ 245--261 \\
35: G. H. Hardy and S. Ramanujan / The normal number of
prime factors of a number $n$ [Quart. J. Math. 48
(1917), 76--92] / 262--275 \\
36: G. H. Hardy and S. Ramanujan / Asymptotic
formul{\ae} in combinatory analysis [Proc. London Math.
Soc. (2) 17 (1918), 75--115] / 276--309 \\
37: G. H. Hardy and S. Ramanujan / On the coefficients
in the expansions of certain modular functions [Proc.
Roy. Soc. A 95 (1919), 144--155] / 310--321 \\
Questions and solutions / 322--334 \\
Appendix I: Notes on the Papers / 335--348 \\
Appendix II: Further Extracts from Ramanujan's Letter
to G. H. Hardy / 349--356 \\
Commentary on Ramanujan's Collected Papers / Bruce C.
Berndt / 357--??",
}
@Article{Jaditz:2000:DPI,
author = "Ted Jaditz",
title = "Are the Digits of $ \pi $ an Independent and
Identically Distributed Sequence?",
journal = j-AMER-STAT,
volume = "54",
number = "1",
pages = "12--16",
month = feb,
year = "2000",
CODEN = "ASTAAJ",
ISSN = "0003-1305 (print), 1537-2731 (electronic)",
ISSN-L = "0003-1305",
bibdate = "Fri Jan 27 18:16:34 MST 2012",
bibsource = "http://www.amstat.org/publications/tas/2000/;
http://www.jstor.org/journals/00031305.html;
http://www.jstor.org/stable/i326507;
https://www.math.utah.edu/pub/tex/bib/amstat2000.bib;
https://www.math.utah.edu/pub/tex/bib/pi.bib",
URL = "http://www.jstor.org/stable/2685604",
acknowledgement = ack-nhfb,
fjournal = "The American Statistician",
journal-URL = "http://www.tandfonline.com/loi/utas20",
xxtitle = "Are the Digits of Pi an iid Sequence?",
}
@Article{Kalantari:2000:NFA,
author = "Bahman Kalantari",
title = "New formulas for approximation of $ \pi $ and other
transcendental numbers",
journal = j-NUMER-ALGORITHMS,
volume = "24",
number = "1--2",
pages = "59--81",
month = dec,
year = "2000",
CODEN = "NUALEG",
ISSN = "1017-1398 (print), 1572-9265 (electronic)",
ISSN-L = "1017-1398",
MRclass = "11J04",
MRnumber = "2001h:11087",
MRreviewer = "David Bradley",
bibdate = "Mon Sep 29 08:37:03 MDT 2003",
bibsource = "http://www.kluweronline.com/issn/1017-1398;
https://www.math.utah.edu/pub/tex/bib/pi.bib; MathSciNet
database",
note = "Computational methods from rational approximation
theory (Wilrijk, 1999).",
URL = "http://ipsapp007.kluweronline.com/content/getfile/5058/27/5/abstract.htm;
http://ipsapp007.kluweronline.com/content/getfile/5058/27/5/fulltext.pdf",
acknowledgement = ack-nhfb,
fjournal = "Numerical Algorithms",
journal-URL = "http://link.springer.com/journal/11075",
}
@Unpublished{Lagarias:2000:NAC,
author = "Jeffrey C. Lagarias",
title = "On the Normality of Arithmetical Constants",
month = sep,
year = "2000",
bibdate = "Sat Apr 23 09:15:29 2011",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
note = "Where is this document?",
acknowledgement = ack-nhfb,
}
@Unpublished{Percival:2000:PDE,
author = "C. Percival",
title = "{PiHex}: a distributed effort to calculate {Pi}",
year = "2000",
bibdate = "Tue Apr 26 09:51:04 2011",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
note = "The computation took two years, and used 250 CPU
years, using otherwise-idle time on 1734 machines in 56
countries.",
URL = "http://oldweb.cecm.sfu.ca/projects/pihex",
acknowledgement = ack-nhfb,
remark = "This now-completed project computed the five
trillionth bit of pi as '0' (starting at bit
4,999,999,999,997: 0x07E45733CC790B5B5979) (1998), the
forty trillionth bit of pi as '0' (starting at bit
39,999,999,999,997: 0xA0F9FF371D17593E0\ldots{})
(1998--1999), and the quadrillionth bit of Pi as '0'
(starting at bit 999,999,999,999,997:
0xE6216B069CB6C1D3) (1998--2000).",
}
@Book{Venkatachala:2000:RP,
editor = "B. J. Venkatachala and V. Vinay and C. S. Yogananda",
title = "{Ramanujan}'s papers",
publisher = "Prism Books",
address = "Bangalore, India",
pages = "391",
year = "2000",
ISBN = "81-7286-180-X",
ISBN-13 = "978-81-7286-180-3",
LCCN = "PN4874.R23 A25 2000",
bibdate = "Fri Jan 9 12:55:00 MST 2015",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib;
z3950.loc.gov:7090/Voyager",
abstract = "Collected papers of Srinivasa Ramanujan Aiyangar,
1887--1920, an Indian mathematician.",
acknowledgement = ack-nhfb,
author-dates = "1887--1920",
subject = "Ramanujan Aiyangar, Srinivasa; Mathematicians",
}
@Article{Xu:2000:C,
author = "De Yi Xu",
title = "The computations of {$ \pi $}",
journal = "J. Central China Normal Univ. Natur. Sci.",
volume = "34",
number = "3",
pages = "376--378",
year = "2000",
CODEN = "HDZKEL",
ISSN = "1000-1190",
MRclass = "11Y60 (01A99)",
MRnumber = "1796020 (2001k:11268)",
bibdate = "Mon Apr 25 16:20:53 2011",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
acknowledgement = ack-nhfb,
fjournal = "Journal of Central China Normal University. Natural
Sciences. Huazhong Shifan Daxue Xuebao. Ziran Kexue
Ban",
}
@Book{Arndt:2001:PU,
author = "J{\"o}rg Arndt and Christoph Haenel",
title = "Pi --- Unleashed",
publisher = pub-SV,
address = pub-SV:adr,
pages = "xii + 270",
year = "2001",
ISBN = "3-540-66572-2 (paperback), 3-642-56735-5 (e-book)",
ISBN-13 = "978-3-540-66572-4 (paperback), 978-3-642-56735-3
(e-book)",
LCCN = "QA484.A7513 2001",
bibdate = "Sat Apr 20 11:01:28 2002",
bibsource = "https://www.math.utah.edu/pub/bibnet/authors/b/borwein-jonathan-m.bib;
https://www.math.utah.edu/pub/tex/bib/master.bib;
https://www.math.utah.edu/pub/tex/bib/pi.bib",
note = "Includes CD-ROM. Translated from the German by
Catriona and David Lischka.",
price = "US\$",
abstract = "Never in the 4000-year history of research into pi
have results been so prolific as at present. In their
book Joerg Arndt and Christoph Haenel describe in
easy-to-understand language the latest and most
fascinating findings of mathematicians and computer
scientists in the field of pi. Attention is focused on
new methods of computation whose speed outstrips that
of predecessor methods by orders of magnitude. The book
comes with a CD-ROM containing not only the source code
of all programs described, but also related texts and
even complete libraries.",
acknowledgement = ack-nhfb,
tableofcontents = "1: The state of Pi art / 1 \\
2: How random is $\pi$? / 21 \\
3: Shortcuts to $\pi$ / 35 \\
4: Approximations for $\pi$ and continued fractions /
51 \\
5: Arcus tangens / 69 \\
6: Spigot algorithms / 77 \\
7: Gauss and $\pi$ / 87 \\
8: Ramanujan and $\pi$ / 103 \\
9: The Borweins and $\pi$ / 113 \\
10: The BBP algorithm / 117 \\
11: Arithmetic / 131 \\
12: Miscellaneous / 153 \\
13: The history of $\pi$ / 165 \\
14: Historical notes / 209 \\
15: The future: $\pi$ calculations on the Internet /
215 \\
16: $\pi$ formula collection / 223 \\
17: Tables / 239 \\
A: Documentation for the {\tt hfloat} Library / 247 \\
Bibliography / 257 \\
Index / 265",
}
@Article{Bailey:2001:PIR,
author = "David H. Bailey and David J. Broadhurst",
title = "Parallel integer relation detection: {Techniques} and
applications",
journal = j-MATH-COMPUT,
volume = "70",
number = "236",
pages = "1719--1736",
month = oct,
year = "2001",
CODEN = "MCMPAF",
ISSN = "0025-5718 (print), 1088-6842 (electronic)",
ISSN-L = "0025-5718",
bibdate = "Mon Jul 16 07:53:14 MDT 2001",
bibsource = "http://www.ams.org/mcom/2001-70-236;
https://www.math.utah.edu/pub/tex/bib/pi.bib",
URL = "http://www.ams.org/journal-getitem?pii=S0025-5718-00-01278-3;
http://www.ams.org/mcom/2001-70-236/S0025-5718-00-01278-3/S0025-5718-00-01278-3.dvi;
http://www.ams.org/mcom/2001-70-236/S0025-5718-00-01278-3/S0025-5718-00-01278-3.pdf;
http://www.ams.org/mcom/2001-70-236/S0025-5718-00-01278-3/S0025-5718-00-01278-3.ps;
http://www.ams.org/mcom/2001-70-236/S0025-5718-00-01278-3/S0025-5718-00-01278-3.tex",
acknowledgement = ack-nhfb,
fjournal = "Mathematics of Computation",
journal-URL = "http://www.ams.org/mcom/",
ORCID-numbers = "Bailey, David H./0000-0002-7574-8342",
}
@Article{Bailey:2001:RCF,
author = "David H. Bailey and Richard E. Crandall",
title = "On the Random Character of Fundamental Constant
Expansions",
journal = j-EXP-MATH,
volume = "10",
number = "2",
pages = "175--190",
month = jun,
year = "2001",
ISSN = "1058-6458 (print), 1944-950X (electronic)",
ISSN-L = "1058-6458",
bibdate = "Sat Apr 23 09:41:21 2011",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
acknowledgement = ack-nhfb,
fjournal = "Experimental mathematics",
journal-URL = "http://www.tandfonline.com/loi/uexm20",
ORCID-numbers = "Bailey, David H./0000-0002-7574-8342",
}
@Misc{OConner:2001:TA,
author = "J. O'Conner and E. F. Robertson",
title = "$ \pi $ through the ages",
howpublished = "Web site.",
year = "2001",
bibdate = "Fri Jul 01 06:46:32 2011",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
URL = "http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Pi_through_the_ages.html",
acknowledgement = ack-nhfb,
}
@Article{Peterson:2001:PMM,
author = "Ivars Peterson",
title = "Pi {\`a} la Mode: Mathematicians tackled the seeming
randomness of pi's digits",
journal = j-SCIENCE-NEWS,
volume = "160",
number = "9",
pages = "136--137",
day = "1",
month = sep,
year = "2001",
CODEN = "SCNEBK",
ISSN = "0036-8423 (print), 1943-0930 (electronic)",
ISSN-L = "0036-8423",
bibdate = "Sat Mar 03 15:27:13 2012",
bibsource = "https://www.math.utah.edu/pub/bibnet/authors/c/crandall-richard-e.bib;
https://www.math.utah.edu/pub/tex/bib/pi.bib;
https://www.math.utah.edu/pub/tex/bib/prng.bib",
URL = "http://www.jstor.org/stable/4012633",
acknowledgement = ack-nhfb,
fjournal = "Science News (Washington, DC)",
keywords = "Richard E. Crandall",
remark = "See \cite{Bailey:2001:RCF} for the research discussed
by Peterson.",
}
@Article{Preuss:2001:DPR,
author = "Paul Preuss",
title = "Are the Digits of Pi Random? {A} {Berkeley Lab}
Researcher May Hold the Key",
journal = "Energy Science News",
volume = "??",
number = "??",
pages = "??--??",
month = "????",
year = "2001",
DOI = "????",
bibdate = "Tue Mar 19 09:52:32 2013",
bibsource = "https://www.math.utah.edu/pub/bibnet/authors/c/crandall-richard-e.bib;
https://www.math.utah.edu/pub/tex/bib/pi.bib",
note = "pnl.gov",
URL = "http://web.archive.org/web/20050208141708/http://www.pnl.gov/energyscience/10-01/art3.htm;
http://www.pnl.gov/energyscience/10-01/art3.htm",
acknowledgement = ack-nhfb,
keywords = "David H. Bailey; Richard E. Crandall",
xxnote = "URL at pnl.gov cannot be found on 19 March 2013;
archive.org has it.",
}
@Article{Bailey:2002:RGN,
author = "David H. Bailey and Richard E. Crandall",
title = "Random Generators and Normal Numbers",
journal = j-EXP-MATH,
volume = "11",
number = "4",
pages = "527--546",
month = "????",
year = "2002",
ISSN = "1058-6458 (print), 1944-950X (electronic)",
ISSN-L = "1058-6458",
bibdate = "Sat Apr 23 09:42:27 2011",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
acknowledgement = ack-nhfb,
fjournal = "Experimental mathematics",
journal-URL = "http://www.tandfonline.com/loi/uexm20",
ORCID-numbers = "Bailey, David H./0000-0002-7574-8342",
}
@Article{Barcenas:2002:CMT,
author = "Di{\'o}medes B{\'a}rcenas and Olga Porras",
title = "Calculation of {$ \pi $} by mean of trigonometric
functions",
journal = "Divulg. Mat.",
volume = "10",
number = "2",
pages = "149--159",
year = "2002",
ISSN = "1315-2068",
MRclass = "11Y60",
MRnumber = "1946906 (2003i:11185)",
MRreviewer = "Duncan A. Buell",
bibdate = "Mon Apr 25 16:00:23 2011",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib;
MathSciNet database",
acknowledgement = ack-nhfb,
fjournal = "Revista Matem{\'a}tica de la Universidad del Zulia.
Divulgaciones Matem{\'a}ticas",
}
@Article{Hertling:2002:SNN,
author = "Peter Hertling",
title = "Simply Normal Numbers to Different Bases",
journal = j-J-UCS,
volume = "8",
number = "2",
pages = "235--242",
day = "28",
month = feb,
year = "2002",
CODEN = "????",
DOI = "https://doi.org/10.3217/jucs-008-02-0235",
ISSN = "0948-695X (print), 0948-6968 (electronic)",
ISSN-L = "0948-6968",
bibdate = "Tue Dec 16 10:06:04 MST 2003",
bibsource = "https://www.math.utah.edu/pub/tex/bib/jucs.bib;
https://www.math.utah.edu/pub/tex/bib/pi.bib;
https://www.math.utah.edu/pub/tex/bib/prng.bib",
URL = "http://www.jucs.org/jucs_8_2/simply_normal_numbers_to",
abstract = "Let $ b \geq 2 $ be an integer. A real number is
called simply normal to base $b$ if in its
representation to base $b$ every digit appears with the
same asymptotic frequency. We answer the following
question for arbitrary integers $a$, $ b \geq 2$: if a
real number is simply normal to base $a$, does this
imply that it is also simply normal to base $b$ ? It
turns out that the answer is different from the well
known answers to the corresponding questions for the
related properties ``normality'', ``disjunctiveness'',
and ``randomness''.",
acknowledgement = ack-nhfb,
fjournal = "J.UCS: Journal of Universal Computer Science",
journal-URL = "http://www.jucs.org/jucs",
keywords = "invariance properties; randomness",
}
@Article{Reid-Green:2002:TEA,
author = "Keith S. Reid-Green",
title = "Three early algorithms: [{Bresenham}'s line-drawing
algorithm; a square-root algorithm; {Machin}'s
algorithm: computation of $ \pi $]",
journal = j-IEEE-ANN-HIST-COMPUT,
volume = "24",
number = "4",
pages = "10--13",
month = oct,
year = "2002",
CODEN = "IAHCEX",
DOI = "https://doi.org/10.1109/MAHC.2002.1114866",
ISSN = "1058-6180 (print), 1934-1547 (electronic)",
ISSN-L = "1058-6180",
bibdate = "Sat Nov 29 16:19:45 MST 2003",
bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib;
https://www.math.utah.edu/pub/tex/bib/ieeeannhistcomput.bib;
https://www.math.utah.edu/pub/tex/bib/pi.bib",
URL = "http://csdl.computer.org/dl/mags/an/2002/04/a4010.htm;
http://csdl.computer.org/dl/mags/an/2002/04/a4010.pdf;
http://csdl.computer.org/dl/mags/an/2002/04/a4010abs.htm",
acknowledgement = ack-nhfb,
fjournal = "IEEE Annals of the History of Computing",
journal-URL = "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=85",
}
@Article{Almkvist:2003:SNF,
author = "Gert Almkvist and Christian Krattenthaler and Joakim
Petersson",
title = "Some New Formulas for $ \pi $",
journal = j-EXP-MATH,
volume = "12",
number = "4",
pages = "441--456",
month = "????",
year = "2003",
CODEN = "????",
ISSN = "1058-6458 (print), 1944-950X (electronic)",
ISSN-L = "1058-6458",
bibdate = "Mon Mar 5 10:25:58 MST 2012",
bibsource = "http://projecteuclid.org/euclid.em;
https://www.math.utah.edu/pub/tex/bib/expmath.bib;
https://www.math.utah.edu/pub/tex/bib/pi.bib",
URL = "http://projecteuclid.org/euclid.em/1087568020",
abstract = "We show how to find series expansions for $ \pi $ of
the form $ \pi = \sum_{n = 0}^\infty S(n) \big / \binom
{mn}{pn}a^n $, where $ S(n) $ is some polynomial in $n$
(depending on $ m, p, a$). We prove that there exist
such expansions for $ m = 8 k$, $ p = 4 k$, $ a = ( -
4)^k$, for any $k$, and give explicit examples for such
expansions for small values of $m$, $p$, and $a$.",
acknowledgement = ack-nhfb,
fjournal = "Experimental Mathematics",
journal-URL = "http://www.tandfonline.com/loi/uexm20",
}
@Book{Borwein:2003:EMC,
author = "Jonathan M. Borwein and David H. Bailey and Roland
Girgensohn",
title = "Experimentation in mathematics: computational paths to
discovery",
publisher = pub-A-K-PETERS,
address = pub-A-K-PETERS:adr,
pages = "x + 357",
year = "2003",
ISBN = "1-56881-136-5",
ISBN-13 = "978-1-56881-136-9",
LCCN = "QA12 .B67 2004",
bibdate = "Mon Feb 07 16:10:50 2005",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
price = "US\$49.00",
acknowledgement = ack-nhfb,
author-dates = "Jonathan Michael Borwein (20 May 1951--2 August
2016)",
ORCID-numbers = "Bailey, David H./0000-0002-7574-8342; Borwein,
Jonathan/0000-0002-1263-0646",
}
@Book{Finch:2003:MC,
author = "Steven R. Finch",
title = "Mathematical Constants",
volume = "94",
publisher = pub-CAMBRIDGE,
address = pub-CAMBRIDGE:adr,
pages = "xix + 602",
year = "2003",
ISBN = "0-521-81805-2 (hardcover), 1-107-26335-2 (e-book),
1-107-26691-2 (e-book)",
ISBN-13 = "978-0-521-81805-6 (hardcover), 978-1-107-26335-2
(e-book), 978-1-107-26691-9 (e-book)",
LCCN = "QA41 .F54 2003",
bibdate = "Mon Dec 31 07:47:16 MST 2007",
bibsource = "https://www.math.utah.edu/pub/tex/bib/master.bib;
https://www.math.utah.edu/pub/tex/bib/pi.bib;
z3950.loc.gov:7090/Voyager",
series = "Encyclopedia of mathematics and its applications",
URL = "http://algo.inria.fr/bsolve/constant/table.html;
http://numbers.computation.free.fr/Constants/constants.html;
http://www.cambridge.org/us/catalogue/catalogue.asp?isbn=0521818052;
http://www.loc.gov/catdir/description/cam031/2002074058.html;
http://www.loc.gov/catdir/samples/cam034/2002074058.html;
http://www.loc.gov/catdir/toc/cam031/2002074058.html",
abstract = "Famous mathematical constants include the ratio of
circular circumference to diameter, $ \pi = 3.14 \ldots
{} $, and the natural logarithmic base, $ e = 2.178
\ldots {} $. Students and professionals usually can
name at most a few others, but there are many more
buried in the literature and awaiting discovery. How do
such constants arise, and why are they important? Here
Steven Finch provides 136 essays, each devoted to a
mathematical constant or a class of constants, from the
well known to the highly exotic. Topics covered include
the statistics of continued fractions, chaos in
nonlinear systems, prime numbers, sum-free sets,
isoperimetric problems, approximation theory,
self-avoiding walks and the Ising model (from
statistical physics), binary and digital search trees
(from theoretical computer science), the
Prouhet--Thue--Morse sequence, complex analysis,
geometric probability and the traveling salesman
problem. This book will be helpful both to readers
seeking information about a specific constant, and to
readers who desire a panoramic view of all constants
coming from a particular field, for example
combinatorial enumeration or geometric optimization.
Unsolved problems appear virtually everywhere as well.
This is an outstanding scholarly attempt to bring
together all significant mathematical constants in one
place.",
acknowledgement = ack-nhfb,
libnote = "Not in my library.",
subject = "Mathematical constants",
tableofcontents = "Volume 1 \\
1. Well-known constants \\
2: Constants associated with number theory \\
3: Constants associated with analytic inequalities \\
4: Constants associated with the approximation of
functions \\
5: Constants associated with enumerating discrete
structures \\
6: Constants associated with functional iteration \\
7: Constants associated with complex analysis \\
8: Constants associated with geometry \\
Volume 2:\\
1: Number theory and combinatorics \\
2: Inequalitites and approximation \\
3: Real and complex analysis \\
4: Probability and stochastic processes \\
5: Geometry and topology",
}
@Article{Gibbs:2003:DSP,
author = "W. W. Gibbs",
title = "A Digital Slice of Pi. The New Way to do Pure Math:
Experimentally",
journal = j-SCI-AMER,
volume = "288",
number = "5",
pages = "23--24",
month = may,
year = "2003",
CODEN = "SCAMAC",
DOI = "https://doi.org/10.1038/scientificamerican0503-23",
ISSN = "0036-8733 (print), 1946-7087 (electronic)",
ISSN-L = "0036-8733",
bibdate = "Tue Apr 26 16:23:52 2011",
bibsource = "https://www.math.utah.edu/pub/bibnet/authors/c/crandall-richard-e.bib;
https://www.math.utah.edu/pub/tex/bib/pi.bib",
URL = "http://crd.lbl.gov/~dhbailey/sciam-2003.pdf;
http://www.nature.com/scientificamerican/journal/v288/n5/pdf/scientificamerican0503-23.pdf;
http://www.scientificamerican.com/article.cfm?id=a-digital-slice-of-pi",
acknowledgement = ack-nhfb,
fjournal = "Scientific American",
journal-URL = "http://www.nature.com/scientificamerican",
keywords = "Richard E. Crandall",
}
@Article{Osmova:2003:CWE,
author = "E. N. Os{\cprime}mova",
title = "Calculation of $ \pi $ in the works of {L. Euler}
using asymptotic series",
journal = "Istor.-Mat. Issled. (2)",
volume = "8(43)",
pages = "167--185, 406",
year = "2003",
ISBN = "5-8037-0160-2",
ISBN-13 = "978-5-8037-0160-6",
MRclass = "01A50",
MRnumber = "2299071",
bibdate = "Mon Apr 25 16:00:23 2011",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib;
MathSciNet database",
ZMnumber = "1179.01013",
acknowledgement = ack-nhfb,
fjournal = "Istoriko-Matematicheskie Issledovaniya. Vtoraya
Seriya",
language = "Russian",
xxtitle = "{Euler}'s calculation of $\pi$ by using an asymptotic
series",
}
@Article{Bailey:2004:BEA,
author = "David H. Bailey and Jonathan M. Borwein and Richard E.
Crandall and Carl Pomerance",
title = "On the Binary Expansions of Algebraic Numbers",
journal = "Journal of Number Theory {Bordeaux}",
volume = "16",
number = "??",
pages = "487--518",
month = "????",
year = "2004",
CODEN = "????",
ISSN = "????",
bibdate = "Sat Apr 23 09:39:50 2011",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
acknowledgement = ack-nhfb,
author-dates = "Jonathan Michael Borwein (20 May 1951--2 August
2016)",
fjournal = "Journal of Number Theory {Bordeaux}",
ORCID-numbers = "Bailey, David H./0000-0002-7574-8342; Borwein,
Jonathan/0000-0002-1263-0646",
}
@Article{Borwein:2004:FEA,
author = "Jonathan M. Borwein and William F. Galway and David
Borwein",
title = "Finding and Excluding $b$-ary {Machin}-Type {BBP}
Formulae",
journal = j-CAN-J-MATH,
volume = "56",
number = "??",
pages = "1339--1342",
month = "????",
year = "2004",
CODEN = "CJMAAB",
ISSN = "0008-414X (print), 1496-4279 (electronic)",
bibdate = "Sat Apr 23 09:12:32 2011",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
acknowledgement = ack-nhfb,
author-dates = "Jonathan Michael Borwein (20 May 1951--2 August
2016)",
fjournal = "Canadian Journal of Mathematics = Journal canadien de
math{\'e}matiques",
journal-URL = "http://cms.math.ca/cjm/",
ORCID-numbers = "Borwein, Jonathan/0000-0002-1263-0646",
remark = "This paper established the result that there are no
degree-1 BBP-type formulas for $ \pi $, except when the
base is 2 (or an integer power thereof).",
}
@Book{Borwein:2004:MEP,
author = "Jonathan M. Borwein and David H. Bailey",
title = "Mathematics by Experiment: Plausible Reasoning in the
{21st Century}",
publisher = pub-A-K-PETERS,
address = pub-A-K-PETERS:adr,
pages = "x + 288",
year = "2004",
ISBN = "1-56881-211-6",
ISBN-13 = "978-1-56881-211-3",
LCCN = "QA76.95 .B67 2003",
bibdate = "Fri Oct 17 10:38:25 2003",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
price = "US\$45.00",
acknowledgement = ack-nhfb,
author-dates = "Jonathan Michael Borwein (20 May 1951--2 August
2016)",
ORCID-numbers = "Bailey, David H./0000-0002-7574-8342; Borwein,
Jonathan/0000-0002-1263-0646",
remark = "Due to an unfortunate error, some of the citations in
the book point to the wrong item in the Bibliography.
Here is how to find the correct citation number:
[1]--[85]: Citation number is correct; [86, page 100]:
[86]; [86, page 2]: [87]; [87]--[156]: Add one to
citation number; [157]: [159]; [158, page 139]: [158];
[158, page 97]: [160]; [159]--[196]: Add two to
citation number",
}
@Article{Byatt:2004:SPD,
author = "D. Byatt and M. L. Dalrymple and R. M. Turner",
title = "Searching for primes in the digits of $ \pi $",
journal = j-COMPUT-MATH-APPL,
volume = "48",
number = "3--4",
pages = "497--504",
month = aug,
year = "2004",
CODEN = "CMAPDK",
ISSN = "0898-1221 (print), 1873-7668 (electronic)",
ISSN-L = "0898-1221",
bibdate = "Wed Mar 1 21:49:39 MST 2017",
bibsource = "https://www.math.utah.edu/pub/tex/bib/computmathappl2000.bib;
https://www.math.utah.edu/pub/tex/bib/pi.bib",
URL = "http://www.sciencedirect.com/science/article/pii/S0898122104840719",
acknowledgement = ack-nhfb,
fjournal = "Computers and Mathematics with Applications",
journal-URL = "http://www.sciencedirect.com/science/journal/08981221",
}
@Book{Eymard:2004:N,
author = "Pierre Eymard and Jean-Pierre Lafon",
title = "The Number $ \pi $",
publisher = pub-AMS,
address = pub-AMS:adr,
pages = "x + 322",
year = "2004",
ISBN = "0-8218-3246-8",
ISBN-13 = "978-0-8218-3246-2",
LCCN = "QA484 .E9613 2004",
bibdate = "Fri Apr 02 14:56:15 2004",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
note = "Translated by Stephen S. Wilson from the French {\em
Autour du nombre $ \pi $} (1999).",
price = "US\$36.00",
URL = "http://www.ams.org/bookpages/tnp/",
acknowledgement = ack-nhfb,
}
@Book{Posamentier:2004:PBW,
author = "Alfred S. Posamentier and Ingmar Lehmann",
title = "$ \pi $: {A} biography of the world's most mysterious
number",
publisher = pub-PROMETHEUS-BOOKS,
address = pub-PROMETHEUS-BOOKS:adr,
pages = "324",
year = "2004",
ISBN = "1-59102-200-2 (hardcover)",
ISBN-13 = "978-1-59102-200-8 (hardcover)",
LCCN = "QA484 .P67 2004",
bibdate = "Sun Feb 17 10:24:30 MST 2013",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib;
z3950.loc.gov:7090/Voyager",
note = "Afterword by Herbert A. Hauptman.",
acknowledgement = ack-nhfb,
subject = "Pi",
}
@Article{Bailey:2005:HPF,
author = "David H. Bailey",
title = "High-Precision Floating-Point Arithmetic in Scientific
Computation",
journal = j-COMPUT-SCI-ENG,
volume = "7",
number = "3",
pages = "54--61",
month = may # "\slash " # jun,
year = "2005",
CODEN = "CSENFA",
DOI = "https://doi.org/10.1109/MCSE.2005.52",
ISSN = "1521-9615 (print), 1558-366X (electronic)",
ISSN-L = "1521-9615",
bibdate = "Sat May 14 13:11:45 MDT 2005",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
URL = "http://csdl.computer.org/comp/mags/cs/2005/03/c3054abs.htm;
http://csdl.computer.org/dl/mags/cs/2005/03/c3054.pdf",
acknowledgement = ack-nhfb,
fjournal = "Computing in Science and Engineering",
journal-URL = "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=5992",
ORCID-numbers = "Bailey, David H./0000-0002-7574-8342",
}
@Article{Chua:2005:EML,
author = "Kok Seng Chua",
title = "Extremal modular lattices, {McKay Thompson} series,
quadratic iterations, and new series for $ \pi $",
journal = j-EXP-MATH,
volume = "14",
number = "3",
pages = "343--357",
month = "????",
year = "2005",
CODEN = "????",
DOI = "https://doi.org/10.1080/10586458.2005.10128932",
ISSN = "1058-6458 (print), 1944-950X (electronic)",
ISSN-L = "1058-6458",
bibdate = "Mon Mar 5 15:33:58 MST 2012",
bibsource = "http://projecteuclid.org/euclid.em;
https://www.math.utah.edu/pub/tex/bib/expmath.bib;
https://www.math.utah.edu/pub/tex/bib/pi.bib;
http://www.tandfonline.com/toc/uexm20/14/3",
URL = "http://projecteuclid.org/euclid.em/1128371759",
acknowledgement = ack-nhfb,
fjournal = "Experimental Mathematics",
journal-URL = "http://www.tandfonline.com/loi/uexm20",
onlinedate = "30 Jan 2011",
}
@Article{Dodge:2005:RNG,
author = "Yadolah Dodge and Giuseppe Melfi",
title = "Random number generators and rare events in the
continued fraction of $ \pi $",
journal = j-J-STAT-COMPUT-SIMUL,
volume = "75",
number = "3",
pages = "189--197",
year = "2005",
CODEN = "JSCSAJ",
DOI = "https://doi.org/10.1080/00949650410001687181",
ISSN = "0094-9655 (print), 1026-7778 (electronic), 1563-5163",
ISSN-L = "0094-9655",
bibdate = "Tue Apr 22 09:12:26 MDT 2014",
bibsource = "https://www.math.utah.edu/pub/tex/bib/jstatcomputsimul.bib;
https://www.math.utah.edu/pub/tex/bib/pi.bib;
https://www.math.utah.edu/pub/tex/bib/prng.bib",
URL = "http://www.tandfonline.com/doi/abs/10.1080/00949650410001687181",
abstract = "Failure of pseudo-random number generators in
producing reliable random numbers as described by Knuth
(Knuth, D. E., 1981, The Art of Computer Programming,
Vol. 2, Addison-Wesley) gave birth to a new generation
of random number generators such as billions of
decimals of $ \pi $. To show that these decimals
satisfy all criterion of being random, Bailey and
Crandall (Bailey, D. B. and Crandall, R. E., 2003,
Random generators and normal numbers, to appear in
Experimental Mathematics) provided a proof toward the
normality of $ \pi $.\par
In this article, we try to show similar results by
considering the continued fraction of $ \pi $, which
appears random as opposed to other supposed normal
numbers whose continued fractions are not random at
all. For this purpose, we analyze the continued
fraction of $ \pi $ and discuss the randomness of its
partial quotients. Some statistical tests are performed
to check whether partial quotients follow the Khinchin
distribution. Finally, we discuss rare elements in the
continued fraction of $ \pi $.",
acknowledgement = ack-nhfb,
fjournal = "Journal of Statistical Computation and Simulation",
journal-URL = "http://www.tandfonline.com/loi/gscs20",
onlinedate = "11 Oct 2011",
}
@Article{Marsaglia:2005:RPO,
author = "George Marsaglia",
title = "On the Randomness of Pi and Other Decimal Expansions",
journal = "{InterStat}: statistics on the {Internet}",
pages = "17",
month = oct,
year = "2005",
CODEN = "????",
ISSN = "1941-689X",
bibdate = "Wed Jun 22 10:34:43 2011",
bibsource = "https://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib;
https://www.math.utah.edu/pub/tex/bib/pi.bib",
URL = "http://interstat.statjournals.net/INDEX/Oct05.html;
http://interstat.statjournals.net/YEAR/2005/articles/0510005.pdf",
abstract = "Tests of randomness much more rigorous than the usual
frequency-of-digit counts are applied to the decimal
expansions of $ \pi $, $e$ and $ \sqrt {2}$, using the
Diehard Battery of Tests adapted to base 10 rather than
the original base 2. The first $ 10^9$ digits of $ \pi
$, $e$ and $ \sqrt {2}$ seem to pass the Diehard tests
very well. But so do the decimal expansions of most
rationals $ k / p$ with large primes $p$. Over the
entire set of tests, only the digits of $ \sqrt {2}$
give a questionable result: the monkey test on 5-letter
words. Its significance is discussed in the
text.\par
Three specific $ k / p$ are used for comparison. The
cycles in their decimal expansions are developed in
reverse order by the multiply-with-carry (MWC) method.
They do well in the Diehard tests, as do many fast and
simple MWC RNGs that produce base-$b$ `digits' of the
expansions of $ k / p$ for $ b = 2^{32}$ or $ b =
2^{32} - 1$. Choices of primes $p$ for such MWC RNGs
are discussed, along with comments on their
implementation.",
abstract-2 = "Extensive tests of randomness used to distinguish good
from not-so-good random number generators are applied
to the digits of $\pi$, $e$ and $\sqrt{2}$, as well as
to rationals $k / p$ for large primes $p$. They seem to
pass these tests as well as some of the best RNGs, and
could well serve in their stead if the digits could be
easily and quickly produced in the computer---and they
can, at least for rationals $k / p$. Simple and fast
methods are developed to produce, in reverse order, for
large primes $p$ and general bases $b$, the periodic
cycles of the base-$b$ expansions of $k / p$. Specific
choices provide high quality, fast and simple RNGs with
periods thousands of orders of magnitude greater than
what are currently viewed as the longest. Also included
are historical references to decimal expansions and
their relation to current, often wrong, website
discussions on the randomness of $\pi$.",
acknowledgement = ack-nhfb,
keywords = "Diehard Tests; Pi; Random Number Generators; Tests of
Randomness",
}
@InCollection{Schumer:2004:ECP,
author = "Peter D. Schumer",
title = "Episodes in the Calculation of Pi",
crossref = "Schumer:2004:MJ",
pages = "101--116",
year = "2004",
bibdate = "Sat Sep 10 16:31:26 2016",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
acknowledgement = ack-nhfb,
}
@Article{Tu:2005:SRD,
author = "Shu-Ju Tu and Ephraim Fischbach",
title = "A Study on the Randomness of the Digits of $ \pi $",
journal = j-INT-J-MOD-PHYS-C,
volume = "16",
number = "2",
pages = "281--294",
month = feb,
year = "2005",
CODEN = "IJMPEO",
DOI = "https://doi.org/10.1142/S0129183105007091",
ISSN = "0129-1831 (print), 1793-6586 (electronic)",
bibdate = "Wed Jun 22 11:19:42 2011",
bibsource = "https://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib;
https://www.math.utah.edu/pub/tex/bib/pi.bib;
https://www.math.utah.edu/pub/tex/bib/prng.bib",
note = "The statistical analysis in this work is flawed; see
\cite{Marsaglia:2005:RPO,Marsaglia:2006:RCS}",
URL = "http://www.worldscinet.com/ijmpc/16/1602/S01291831051602.html",
abstract = "We apply a newly-developed computational method,
Geometric Random Inner Products (GRIP), to quantify the
randomness of number sequences obtained from the
decimal digits of $ \pi $. Several members from the
GRIP family of tests are used, and the results from $
\pi $ are compared to those calculated from other
random number generators. These include a recent
hardware generator based on an actual physical process,
turbulent electroconvection. We find that the decimal
digits of $ \pi $ are in fact good candidates for
random number generators and can be used for practical
scientific and engineering computations.",
acknowledgement = ack-nhfb,
fjournal = "International Journal of Modern Physics C [Physics and
Computers]",
journal-URL = "http://www.worldscientific.com/loi/ijmpc",
}
@Book{Adams:2006:GDW,
author = "Colin Conrad Adams and Edward B. Burger and Thomas A.
Garrity",
title = "The great $ \pi $ /$e$ debate: [which is the better
number?]",
publisher = pub-MATH-ASSOC-AMER,
address = pub-MATH-ASSOC-AMER:adr,
year = "2006",
ISBN = "0-88385-900-9",
ISBN-13 = "978-0-88385-900-1",
LCCN = "QA99 .A33 2006 DVD",
bibdate = "Fri Feb 15 10:02:40 MST 2019",
bibsource = "fsz3950.oclc.org:210/WorldCat;
https://www.math.utah.edu/pub/tex/bib/pi.bib",
note = "One 40-minute DVD.",
URL = "https://www.reddit.com/r/math/comments/na7ua/pi_vs_e_debate/",
abstract = "Two professors from Williams College present
arguments, challenge orthodoxy, brazenly flaunt
convention, and often behave badly in this humorous
debate over the relative merits of everyone's favorite
transcendental numbers, pi and e, staged for the 1st
Year Family Weekend at Williams College on October 29,
2005. The debate participants: Colin Adams states the
case for pi, Thomas Garrity is e's champion, and Edward
Burger, chair of the Dept. of Mathematics and
Statistics at Williams College, serves as moderator.",
}
@Article{Boslaugh:2006:BRG,
author = "Sarah Boslaugh",
title = "Book Review: {{\booktitle{The Great Pi/e Debate}},
Colin Adams and Thomas Garrity Mathematical Association
of America, 2006, \$24.95 ISBN 0-88385-900-9}",
journal = "MAA Reviews",
volume = "??",
number = "??",
pages = "??--?",
day = "28",
month = dec,
year = "2006",
bibdate = "Fri Feb 15 10:07:28 2019",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
URL = "https://www.maa.org/press/maa-reviews/the-great-pie-debate",
acknowledgement = ack-nhfb,
}
@Article{Chan:2006:T,
author = "Hei-Chi Chan",
title = "$ \pi $ in terms of $ \phi $",
journal = j-FIB-QUART,
volume = "44",
number = "2",
pages = "141--144",
month = may,
year = "2006",
CODEN = "FIBQAU",
ISSN = "0015-0517",
ISSN-L = "0015-0517",
bibdate = "Thu Oct 20 18:04:12 MDT 2011",
bibsource = "http://www.fq.math.ca/44-2.html;
https://www.math.utah.edu/pub/tex/bib/fibquart.bib;
https://www.math.utah.edu/pub/tex/bib/pi.bib",
URL = "http://www.fq.math.ca/Abstracts/44-2/chan.pdf",
acknowledgement = ack-nhfb,
ajournal = "Fib. Quart",
fjournal = "The Fibonacci Quarterly",
journal-URL = "http://www.fq.math.ca/",
}
@Article{Guillera:2006:CCS,
author = "Jes{\'u}s Guillera",
title = "A Class of Conjectured Series Representations for $ 1
/ \pi $",
journal = j-EXP-MATH,
volume = "15",
number = "4",
pages = "409--414",
month = "????",
year = "2006",
CODEN = "????",
DOI = "https://doi.org/10.1080/10586458.2006.10128971",
ISSN = "1058-6458 (print), 1944-950X (electronic)",
ISSN-L = "1058-6458",
bibdate = "Mon Mar 5 15:43:50 MST 2012",
bibsource = "http://projecteuclid.org/euclid.em;
https://www.math.utah.edu/pub/tex/bib/expmath.bib;
https://www.math.utah.edu/pub/tex/bib/pi.bib;
http://www.tandfonline.com/toc/uexm20/15/4",
URL = "http://projecteuclid.org/euclid.em/1175789776",
acknowledgement = ack-nhfb,
fjournal = "Experimental Mathematics",
journal-URL = "http://www.tandfonline.com/loi/uexm20",
onlinedate = "30 Jan 2011",
}
@Article{Guillera:2006:NMO,
author = "Jes{\'u}s Guillera",
title = "A New Method to Obtain Series for $ 1 / \pi $ and $ 1
/ \pi^2 $",
journal = j-EXP-MATH,
volume = "15",
number = "1",
pages = "83--89",
month = "????",
year = "2006",
CODEN = "????",
DOI = "https://doi.org/10.1080/10586458.2006.10128943",
ISSN = "1058-6458 (print), 1944-950X (electronic)",
ISSN-L = "1058-6458",
bibdate = "Mon Mar 5 15:33:58 MST 2012",
bibsource = "http://projecteuclid.org/euclid.em;
https://www.math.utah.edu/pub/tex/bib/expmath.bib;
https://www.math.utah.edu/pub/tex/bib/pi.bib;
http://www.tandfonline.com/toc/uexm20/15/1",
URL = "http://projecteuclid.org/euclid.em/1150476906",
acknowledgement = ack-nhfb,
fjournal = "Experimental Mathematics",
journal-URL = "http://www.tandfonline.com/loi/uexm20",
onlinedate = "30 Jan 2011",
}
@Article{Marsaglia:2006:RCS,
author = "George Marsaglia",
title = "Refutation of claims such as {``Pi is less random than
we thought''}",
journal = "{InterStat}: statistics on the {Internet}",
day = "23",
month = jan,
year = "2006",
CODEN = "????",
ISSN = "1941-689X",
bibdate = "Tue Jun 21 19:08:05 2011",
bibsource = "https://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib;
https://www.math.utah.edu/pub/tex/bib/pi.bib",
URL = "http://interstat.statjournals.net/YEAR/2006/articles/0601001.pdf",
abstract = "In article by Tu and Fischman in a Physics journal
\cite{Tu:2005:SRD} has led to worldwide reports that Pi
is less random than we thought, or that Pi is not the
best random number generator, or that Pi seems good but
not the best. A careful examination of the Tu and
Fischman procedure shows that it is needlessly
complicated and can be reduced to study of the average
value of $ (U_2 - U_1) (U_2 - U_3) $ for uniform
variates U produced by a RNG, (but not on their
distribution). The authors' method of assigning a
letter grade, A+, A, B, C, D, E to a sample mean, based
on its distance from the expected value, suggests
naivety in the extreme. Application, in the present
article, to the first 960 million digits of the
expansion of Pi shows that they perform as well as
other RNGs on not only the average for $ (U_2 - U_1)
(U_2 - U_3) $, but on the more difficult test for their
distribution, consistent with results previously shown
in this journal that Pi does quite well on far more
extensive and difficult-to-pass tests of randomness.",
acknowledgement = ack-nhfb,
keywords = "Diehard Tests; LSTests of Randomness; Pi; Random
Number Generators",
}
@Book{Bailey:2007:EMA,
author = "David H. Bailey and Jonathan M. Borwein and Neil J.
Calkin and Roland Girgensohn and D. Russell Luke and
Victor Moll",
title = "Experimental Mathematics in Action",
publisher = pub-A-K-PETERS,
address = pub-A-K-PETERS:adr,
pages = "xii + 322",
year = "2007",
ISBN = "1-56881-271-X",
ISBN-13 = "978-1-56881-271-7",
LCCN = "QA8.7 .E97 2007",
bibdate = "Thu Oct 25 18:45:59 MDT 2007",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib;
z3950.loc.gov:7090/Voyager",
acknowledgement = ack-nhfb,
author-dates = "Jonathan Michael Borwein (20 May 1951--2 August
2016)",
ORCID-numbers = "Bailey, David H./0000-0002-7574-8342; Borwein,
Jonathan/0000-0002-1263-0646",
subject = "Experimental mathematics",
}
@Book{Borwein:2008:CMD,
editor = "Jonathan M. Borwein and E. M. (Eugenio M.) Rocha and
Jos{\'e}-Francisco Rodrigues",
title = "Communicating mathematics in the digital era",
publisher = pub-A-K-PETERS,
address = pub-A-K-PETERS:adr,
pages = "xii + 325",
year = "2008",
ISBN = "1-56881-410-0",
ISBN-13 = "978-1-56881-410-0",
LCCN = "QA76.95 .C59 2008",
bibdate = "Tue Nov 10 17:48:02 MST 2009",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib;
z3950.loc.gov:7090/Voyager",
URL = "http://www.loc.gov/catdir/toc/fy0903/2008022183.html",
acknowledgement = ack-nhfb,
libnote = "Not in my library.",
remark = "This book reflects many of the contributions \ldots{}
that were delivered and discussed at the ICM 2006
satellite meeting entitled ``Communicating Mathematics
in the Digital Era'' (CMDE2006), which took place at
the University of Aveiro in Portugal, August 15--18,
2006.",
subject = "mathematics; data processing; congresses; libraries
and electronic publishing; image processing; digital
techniques",
}
@InProceedings{Borwein:2008:VPG,
author = "J. M. Borwein",
editor = "????",
booktitle = "Mathematics and Culture, La matematica: Problemi e
teoremi",
title = "La vita di pi greco. ({Italian}) [{The} life of
{Greek} pi]",
publisher = "Guilio Einaudi Editori",
address = "Turino, Italy",
pages = "??--??",
year = "2008",
ISBN = "????",
ISBN-13 = "????",
LCCN = "????",
bibdate = "Sat Apr 23 09:46:00 2011",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
URL = "http://www.carma.newcastle.edu.au/~jb616/pi-2010.pdf",
acknowledgement = ack-nhfb,
author-dates = "Jonathan Michael Borwein (20 May 1951--2 August
2016)",
language = "Italian",
ORCID-numbers = "Borwein, Jonathan/0000-0002-1263-0646",
}
@Article{Chan:2008:MTF,
author = "Hei-Chi Chan",
title = "{Machin}-Type Formulas Expressing $ \pi $ in Terms of
$ \phi $",
journal = j-FIB-QUART,
volume = "46/47",
number = "1",
pages = "32--37",
month = feb,
year = "2008\slash 2009",
CODEN = "FIBQAU",
ISSN = "0015-0517",
ISSN-L = "0015-0517",
bibdate = "Thu Oct 20 18:04:27 MDT 2011",
bibsource = "http://www.fq.math.ca/46/47-1.html;
https://www.math.utah.edu/pub/tex/bib/fibquart.bib;
https://www.math.utah.edu/pub/tex/bib/pi.bib",
URL = "http://www.fq.math.ca/Abstracts/46_47-1/chan.pdf",
acknowledgement = ack-nhfb,
ajournal = "Fib. Quart",
fjournal = "The Fibonacci Quarterly",
journal-URL = "http://www.fq.math.ca/",
}
@Article{Chong:2008:EQ,
author = "Terence Tai-Leung Chong",
title = "The empirical quest for $ \pi $",
journal = j-COMPUT-MATH-APPL,
volume = "56",
number = "10",
pages = "2772--2778",
month = nov,
year = "2008",
CODEN = "CMAPDK",
DOI = "https://doi.org/10.1016/j.camwa.2008.07.005",
ISSN = "0898-1221 (print), 1873-7668 (electronic)",
ISSN-L = "0898-1221",
bibdate = "Tue Feb 14 09:49:52 2012",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
URL = "http://www.sciencedirect.com/science/article/pii/S0898122108004306",
acknowledgement = ack-nhfb,
fjournal = "Computers and Mathematics with Applications",
remark = "This article describes one of the slowest ways to
compute $ \pi $, from probabilistic estimates using
real-world data!",
}
@Article{Guillera:2008:EPS,
author = "Jes{\'u}s Guillera",
title = "Easy Proofs of Some {Borwein} Algorithms for $ \pi $",
journal = j-AMER-MATH-MONTHLY,
volume = "115",
number = "9",
pages = "850--854",
month = nov,
year = "2008",
CODEN = "AMMYAE",
ISSN = "0002-9890 (print), 1930-0972 (electronic)",
ISSN-L = "0002-9890",
bibdate = "Mon Jan 30 12:00:32 MST 2012",
bibsource = "http://www.jstor.org/journals/00029890.html;
http://www.jstor.org/stable/i27642605;
https://www.math.utah.edu/pub/tex/bib/amermathmonthly2000.bib;
https://www.math.utah.edu/pub/tex/bib/pi.bib",
URL = "http://www.jstor.org/stable/27642614",
acknowledgement = ack-nhfb,
fjournal = "American Mathematical Monthly",
journal-URL = "https://www.jstor.org/journals/00029890.htm",
subject-dates = "Jonathan Michael Borwein (20 May 1951--2 August
2016)",
}
@Article{Hogendijk:2008:AKD,
author = "Jan P. Hogendijk",
title = "{Al-K{\=a}sh{\=\i}}'s determination of $ \pi $ to $ 16
$ decimals in an old manuscript",
journal = "Z. Gesch. Arab.-Islam. Wiss.",
volume = "18",
pages = "73--153",
year = "2008\slash 2009",
ISSN = "0179-4639",
MRclass = "01A30",
MRnumber = "2572309 (2010i:01002)",
bibdate = "Mon Apr 25 16:27:00 2011",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
note = "With an appendix containing Al-K{\=a}sh{\=\i}'s {\it
Treatise on the Circumference} in Arabic.",
acknowledgement = ack-nhfb,
fjournal = "Zeitschrift f{\"u}r Geschichte der
Arabisch-Islamischen Wissenschaften",
}
@Article{Miller:2008:PPW,
author = "Steven J. Miller",
title = "A Probabilistic Proof of {Wallis}'s Formula for $ \pi
$",
journal = j-AMER-MATH-MONTHLY,
volume = "115",
number = "8",
pages = "740--745",
month = oct,
year = "2008",
CODEN = "AMMYAE",
ISSN = "0002-9890 (print), 1930-0972 (electronic)",
ISSN-L = "0002-9890",
bibdate = "Mon Jan 30 12:00:31 MST 2012",
bibsource = "http://www.jstor.org/journals/00029890.html;
http://www.jstor.org/stable/i27642579;
https://www.math.utah.edu/pub/tex/bib/amermathmonthly2000.bib;
https://www.math.utah.edu/pub/tex/bib/pi.bib",
URL = "http://www.jstor.org/stable/27642585",
acknowledgement = ack-nhfb,
fjournal = "American Mathematical Monthly",
journal-URL = "https://www.jstor.org/journals/00029890.htm",
}
@Article{Pickett:2008:ACF,
author = "Thomas J. Pickett and Ann Coleman",
title = "Another continued fraction for {$ \pi $}",
journal = j-AMER-MATH-MONTHLY,
volume = "115",
number = "10",
pages = "930--933",
year = "2008",
CODEN = "AMMYAE",
ISSN = "0002-9890 (print), 1930-0972 (electronic)",
ISSN-L = "0002-9890",
MRclass = "11A55",
MRnumber = "2468553",
bibdate = "Wed Jan 14 13:22:34 2015",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
URL = "http://www.jstor.org/stable/27642639",
acknowledgement = ack-nhfb,
fjournal = "American Mathematical Monthly",
journal-URL = "https://www.jstor.org/journals/00029890.htm",
}
@Article{Pilehrood:2008:ALC,
author = "Kh. Hessami Pilehrood and T. Hessami Pilehrood",
title = "An {Ap{\'e}ry}-like continued fraction for {$ \pi {\rm
coth} \, \pi $}",
journal = j-J-DIFFERENCE-EQU-APPL,
volume = "14",
number = "12",
pages = "1279--1287",
year = "2008",
CODEN = "JDEAEA",
DOI = "https://doi.org/10.1080/10236190801945571",
ISSN = "1023-6198",
ISSN-L = "1023-6198",
MRclass = "11J70 (11B37 33C20 33F10)",
MRnumber = "2462530 (2009k:11119)",
MRreviewer = "Richard T. Bumby",
bibdate = "Wed Jan 14 13:22:34 2015",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
acknowledgement = ack-nhfb,
fjournal = "Journal of Difference Equations and Applications",
journal-URL = "http://www.informaworld.com/1023-6198",
keywords = "continued fraction; hypergeometric series; rational
function; Zeilberger's algorithm",
}
@Misc{Bellard:2009:PFA,
author = "Fabrice Bellard",
title = "Pi Formulas, Algorithms and Computations",
howpublished = "Web site.",
day = "31",
month = dec,
year = "2009",
bibdate = "Wed Dec 05 14:10:48 2018",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
URL = "https://bellard.org/pi/",
acknowledgement = ack-nhfb,
}
@Book{Borwein:2009:CCI,
author = "Jonathan M. Borwein and Keith J. Devlin",
title = "The computer as crucible: an introduction to
experimental mathematics",
publisher = pub-A-K-PETERS,
address = pub-A-K-PETERS:adr,
pages = "xi + 158",
year = "2009",
ISBN = "1-56881-343-0",
ISBN-13 = "978-1-56881-343-1",
LCCN = "QA8.7 .B67 2009",
bibdate = "Tue Nov 10 17:48:24 MST 2009",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib;
z3950.loc.gov:7090/Voyager",
URL = "http://www.loc.gov/catdir/toc/fy0904/2008022180.html",
acknowledgement = ack-nhfb,
author-dates = "Jonathan Michael Borwein (20 May 1951--2 August
2016)",
libnote = "Not in my library.",
ORCID-numbers = "Borwein, Jonathan/0000-0002-1263-0646",
subject = "Experimental mathematics",
tableofcontents = "What is experimental mathematics? \\
What is the quadrillionth decimal place of $pi$? \\
What is that number? \\
The most important function in mathematics \\
Evaluate the following integral \\
Serendipity \\
Calculating [pi] \\
The computer knows more math than you do \\
Take it to the limit \\
Danger! Always exercise caution when using the computer
\\
Stuff we left out (until now)",
}
@Misc{USCongress:2009:HRP,
author = "{United States Congress}",
title = "{House Resolution 224}: Pi day",
howpublished = "Web document",
day = "12",
month = mar,
year = "2009",
bibdate = "Mon Mar 19 10:41:23 2012",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
note = "The resolution ends with: ``Resolved, That the House
of Representatives-- (1) supports the designation of a
``Pi Day'' and its celebration around the world; (2)
recognizes the continuing importance of National
Science Foundation's math and science education
programs; and (3) encourages schools and educators to
observe the day with appropriate activities that teach
students about Pi and engage them about the study of
mathematics.''",
acknowledgement = ack-nhfb,
}
@Unpublished{Adegoke:2010:NBD,
author = "Kunle Adegoke",
title = "New Binary Degree 3 Digit Extraction ({BBP}-type)
Formulas",
month = dec,
year = "2010",
bibdate = "Sat Apr 23 09:17:57 2011",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
note = "Where is this document?",
URL = "http://adegoke.atwebpages.com/",
acknowledgement = ack-nhfb,
}
@Article{Adegoke:2010:NBT,
author = "Kunle Adegoke",
title = "New Binary and Ternary Digit Extraction ({BBP}-type)
Formulas for Trilogarithm Constants",
journal = "New York Journal of Mathematics",
volume = "16",
number = "??",
pages = "361--367",
month = "????",
year = "2010",
bibdate = "Sat Apr 23 09:22:51 2011",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
URL = "http://nyjm.albany.edu/j/2010/16-14v.pdf",
acknowledgement = ack-nhfb,
fjournal = "New York Journal of Mathematics",
}
@Article{Adegoke:2010:NPR,
author = "Kunle Adegoke",
title = "Non-{PSLQ} Route to {BBP}-type Formulas",
journal = "Journal of Mathematics Research",
volume = "2",
number = "2",
pages = "56--64",
month = "????",
year = "2010",
CODEN = "????",
ISSN = "????",
bibdate = "Sat Apr 23 09:21:15 2011",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
URL = "http://www.ccsenet.org/journal/index.php/jmr/article/download/3853/4736",
acknowledgement = ack-nhfb,
fjournal = "Journal of Mathematics Research",
}
@Article{Almkvist:2010:RLS,
author = "Gert Almkvist and Jesus Guillera",
title = "{Ramanujan}-like series for $ 1 / \pi^2 $ and string
theory",
journal = "arxiv.org",
volume = "??",
number = "??",
pages = "??--??",
day = "27",
month = sep,
year = "2010",
bibdate = "Tue Apr 21 16:14:50 2020",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
URL = "http://arxiv.org/abs/1009.5202.",
acknowledgement = ack-nhfb,
}
@Unpublished{Brent:2010:MPZ,
author = "Richard P. Brent",
title = "Multiple-precision zero-finding methods and the
complexity of elementary function evaluation",
day = "20",
month = apr,
year = "2010",
MRclass = "11Y60 (Primary), 65Y20 (Secondary)",
bibdate = "Tue Apr 26 14:13:36 2011",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
note = "Reprint of \cite{Brent:1976:MPZ} with a postscript
describing more recent developments. See also
\cite{Salamin:1976:CUA}",
URL = "http://arxiv.org/abs/1004.3412v2;
http://wwwmaths.anu.edu.au/~brent/pub/pub028.html",
abstract = "We consider methods for finding high-precision
approximations to simple zeros of smooth functions. As
an application, we give fast methods for evaluating the
elementary functions $ \log (x) $, $ \exp (x) $, $ \sin
(x) $ etc. to high precision. For example, if $x$ is a
positive floating-point number with an $n$-bit
fraction, then (under rather weak assumptions) an
$n$-bit approximation to $ \log (x)$ or $ \exp (x)$ may
be computed in time asymptotically equal to $ 13 M(n)
\lg (n)$, where $ M(n)$ is the time required to
multiply floating-point numbers with $n$-bit fractions.
Similar results are given for the other elementary
functions. Some analogies with operations on formal
power series (over a field of characteristic zero) are
discussed. In particular, it is possible to compute the
first $n$ terms in $ \log (1 + a_1 x + \cdots)$ or $
\exp (a_1. x) + \cdots $ in time $ O(M(n))$, where $
M(n)$ is the time required to multiply two polynomials
of degree $ n - 1$. It follows that the first $n$ terms
in a $q$-th power $ (1 + a_1 x + \cdots)^q$ can be
computed in time $ O(M(n))$, independent of $q$. One of
the results of this paper is the ``Gauss--Legendre'' or
``Brent--Salamin'' algorithm for computing pi. This is
the first quadratically convergent algorithm for pi. It
was also published in Brent [J. ACM 23 (1976),
242--251], and independently by Salamin [Math. Comp. 30
(1976), 565--570].",
acknowledgement = ack-nhfb,
}
@Article{Calude:2010:EEQ,
author = "Cristian S. Calude and Michael J. Dinneen and Monica
Dumitrescu and Karl Svozil",
title = "Experimental evidence of quantum randomness
incomputability",
journal = j-PHYS-REV-A,
volume = "82",
number = "2",
pages = "022102",
month = aug,
year = "2010",
CODEN = "PLRAAN",
DOI = "https://doi.org/10.1103/PhysRevA.82.022102",
ISSN = "1050-2947 (print), 1094-1622, 1538-4446, 1538-4519",
ISSN-L = "1050-2947",
bibdate = "Sat Apr 8 10:48:45 2017",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib;
https://www.math.utah.edu/pub/tex/bib/prng.bib",
URL = "http://link.aps.org/doi/10.1103/PhysRevA.82.022102",
fjournal = "Physical Review A (Atomic, Molecular, and Optical
Physics)",
journal-URL = "http://pra.aps.org/browse",
numpages = "8",
remark = "This paper investigates possible differences in
measures of randomness among algorithms for
pseudorandom numbers, quantum random numbers, and the
digits of $ \pi $.",
}
@Article{Jauregui:2010:NRD,
author = "M. Jauregui and C. Tsallis",
title = "New representations of $ \pi $ and {Dirac} delta using
the nonextensive-statistical-mechanics $q$-exponential
function",
journal = j-J-MATH-PHYS,
volume = "51",
number = "6",
pages = "063304",
month = jun,
year = "2010",
CODEN = "JMAPAQ",
DOI = "https://doi.org/10.1063/1.3431981",
ISSN = "0022-2488 (print), 1089-7658 (electronic), 1527-2427",
ISSN-L = "0022-2488",
bibdate = "Wed Oct 26 16:59:50 MDT 2011",
bibsource = "http://www.aip.org/ojs/jmp.html;
https://www.math.utah.edu/pub/tex/bib/pi.bib",
URL = "http://jmp.aip.org/resource/1/jmapaq/v51/i6/p063304_s1",
abstract = "We present a generalization of the representation in
plane waves of Dirac delta, $ \delta (x) = (1 / 2 \pi)
\int_{- \infty }^\infty e^{-ikx} \, d k $, namely, $
\delta (x) = [(2 - q) / 2 \pi] \int_{- \infty }^{\infty
} e_q^{-ikx} \, d k $, using the
non-extensive-statistical-mechanics $q$-exponential
function, $ e_q^{ix} \equiv [1 + (1 - q) i x]^{1 / (1 -
q)}$ with $ e_1^{ix} \equiv e^{ix}$, $x$ being any real
number, for real values of $q$ within the interval $
[1, 2 [$. Concomitantly, with the development of these
new representations of Dirac delta, we also present two
new families of representations of the transcendental
number $ \pi $. Incidentally, we remark that the
$q$-plane wave form which emerges, namely, $
e_q^{ikx}$, is normalizable for $ 1 < q < 3$, in
contrast to the standard one, $ e^{ikx}$, which is
not.",
acknowledgement = ack-nhfb,
fjournal = "Journal of Mathematical Physics",
journal-URL = "http://jmp.aip.org/",
onlinedate = "29 June 2010",
pagecount = "9",
}
@Article{Jones:2010:DPI,
author = "Timothy W. Jones",
title = "Discovering and Proving that $ \pi $ Is Irrational",
journal = j-AMER-MATH-MONTHLY,
volume = "117",
number = "6",
pages = "553--557",
month = jun,
year = "2010",
CODEN = "AMMYAE",
DOI = "https://doi.org/10.4169/000298910X492853",
ISSN = "0002-9890 (print), 1930-0972 (electronic)",
ISSN-L = "0002-9890",
bibdate = "Mon Jan 30 08:58:17 MST 2012",
bibsource = "http://www.jstor.org/journals/00029890.html;
http://www.jstor.org/stable/10.4169/amermathmont.117.issue-6;
https://www.math.utah.edu/pub/tex/bib/amermathmonthly2010.bib;
https://www.math.utah.edu/pub/tex/bib/pi.bib",
URL = "http://www.jstor.org/stable/pdfplus/10.4169/000298910X492853.pdf",
acknowledgement = ack-nhfb,
fjournal = "American Mathematical Monthly",
journal-URL = "https://www.jstor.org/journals/00029890.htm",
}
@Article{Kaneko:2010:NNP,
author = "Hajime Kaneko",
title = "On normal numbers and powers of algebraic numbers",
journal = "Integers",
volume = "10",
pages = "A5, 31--64",
year = "2010",
DOI = "https://doi.org/10.1515/INTEG.2010.005",
ISSN = "1867-0652",
MRclass = "11K16 (11K06)",
MRnumber = "2601309 (2011b:11105)",
MRreviewer = "M. Mend{\`e}s France",
bibdate = "Fri May 3 18:43:41 2013",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
acknowledgement = ack-nhfb,
fjournal = "Integers. Electronic Journal of Combinatorial Number
Theory",
remark = "See \cite[page 377]{Bailey:2012:EAN} for the
significance of this work.",
}
@Article{Osler:2010:LBF,
author = "Thomas J. Osler",
title = "{Lord Brouncker}'s forgotten sequence of continued
fractions for pi",
journal = j-INT-J-MATH-EDU-SCI-TECH,
volume = "41",
number = "1",
pages = "105--110",
year = "2010",
CODEN = "IJMEBM",
DOI = "https://doi.org/10.1080/00207390903189195",
ISSN = "0020-739X",
ISSN-L = "0020-739X",
MRclass = "01A45 (11A55)",
MRnumber = "2786244 (2012g:01010)",
bibdate = "Wed Jan 14 13:22:34 2015",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
acknowledgement = ack-nhfb,
fjournal = "International Journal of Mathematical Education in
Science and Technology",
}
@Article{Sondow:2010:NWC,
author = "Jonathan Sondow and Huang Yi",
title = "New {Wallis}- and {Catalan}-Type Infinite Products for
$ \pi $, $e$ and $ \sqrt {2 + \sqrt {2}}$",
journal = j-AMER-MATH-MONTHLY,
volume = "117",
number = "10",
pages = "912--917",
month = dec,
year = "2010",
CODEN = "AMMYAE",
ISSN = "0002-9890 (print), 1930-0972 (electronic)",
ISSN-L = "0002-9890",
bibdate = "Mon Jan 30 08:58:16 MST 2012",
bibsource = "http://www.jstor.org/journals/00029890.html;
http://www.jstor.org/stable/10.4169/amermathmont.117.issue-10;
https://www.math.utah.edu/pub/tex/bib/amermathmonthly2010.bib;
https://www.math.utah.edu/pub/tex/bib/pi.bib",
URL = "http://www.jstor.org/stable/pdfplus/10.4169/000298910X523399.pdf",
acknowledgement = ack-nhfb,
fjournal = "American Mathematical Monthly",
journal-URL = "https://www.jstor.org/journals/00029890.htm",
}
@InProceedings{Sze:2010:TQB,
author = "Tsz-Wo Sze",
editor = "{IEEE}",
booktitle = "{2010 IEEE Second International Conference on Cloud
Computing Technology and Science (CloudCom)}",
title = "The Two Quadrillionth Bit of Pi is $0$ ! Distributed
Computation of Pi with {Apache Hadoop}",
publisher = pub-IEEE,
address = pub-IEEE:adr,
pages = "727",
year = "2010",
DOI = "https://doi.org/10.1109/CloudCom.2010.57",
ISBN = "1-4244-9405-2",
ISBN-13 = "978-1-4244-9405-7",
LCCN = "????",
bibdate = "Mon Apr 25 18:16:05 2011",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
abstract = "We present a new record on computing specific bits of
Pi, the mathematical constant, and discuss performing
such computations on Apache Hadoop clusters. The
specific bits represented in hexadecimal are 0E6C1294
AED40403 F56D2D76 4026265B CA98511D 0FCFFAA1 0F4D28B1
BB5392B8. These 256 bits end at the
2,000,000,000,000,252nd bit position, which doubles the
position and quadruples the precision of the previous
known record. The position of the first bit is
1,999,999,999,999,997 and the value of the two
quadrillionth bit is 0. The computation is carried out
by a MapReduce program called DistBbp. To effectively
utilize available cluster resources without
monopolizing the whole cluster, we develop an elastic
computation framework that automatically schedules
computation slices, each a DistBbp job, as either
map-side or reduce-side computation based on changing
cluster load condition. We have calculated Pi at
varying bit positions and precisions, and one of the
largest computations took 23 days of wall clock time
and 503 years of CPU time on a 1000-node cluster.",
acknowledgement = ack-nhfb,
remark = "This paper contains a good discussion of
floating-point rounding errors in the BBP algorithm,
and of the optimal way to distribute computations over
multiple independent systems sharing a common
filesystem (needed to permit restart after node
failure).",
}
@Article{Takahashi:2010:PIM,
author = "Daisuke Takahashi",
title = "Parallel implementation of multiple-precision
arithmetic and $ 2, 576, 980, 370, 000 $ decimal digits
of $ \pi $ calculation",
journal = j-PARALLEL-COMPUTING,
volume = "36",
number = "8",
pages = "439--448",
month = aug,
year = "2010",
CODEN = "PACOEJ",
DOI = "https://doi.org/10.1016/j.parco.2010.02.007",
ISSN = "0167-8191 (print), 1872-7336 (electronic)",
ISSN-L = "0167-8191",
bibdate = "Thu Sep 2 17:51:13 MDT 2010",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib;
http://www.sciencedirect.com/science/journal/01678191",
abstract = "We present efficient parallel algorithms for
multiple-precision arithmetic operations of more than
several million decimal digits on distributed-memory
parallel computers. A parallel implementation of
floating-point real FFT-based multiplication is used,
since the key operation for fast multiple-precision
arithmetic is multiplication. The operation for
releasing propagated carries and borrows in
multiple-precision addition, subtraction and
multiplication was also parallelized. More than 2.576
trillion decimal digits of $ \pi $ were computed on 640
nodes of Appro Xtreme-X3 (648 nodes, 147.2 GFlops/node,
95.4 TFlops peak performance) with a computing elapsed
time of 73 h 36 min which includes the time required
for verification.",
acknowledgement = ack-nhfb,
fjournal = "Parallel Computing",
journal-URL = "http://www.sciencedirect.com/science/journal/01678191",
keywords = "distributed-memory parallel computer; Fast Fourier
transform; multiple-precision arithmetic",
}
@Unpublished{Adegoke:2011:CBB,
author = "Kunle Adegoke",
title = "A Class of Binary {BBP}-type Formulas in General
Degrees",
month = feb,
year = "2011",
bibdate = "Sat Apr 23 09:24:34 2011",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
note = "Where is this document?",
URL = "http://adegoke.atwebpages.com/",
acknowledgement = ack-nhfb,
}
@Unpublished{Adegoke:2011:FPD,
author = "Kunle Adegoke",
title = "Formal Proofs of Degree 5 Binary {BBP}-type Formulas",
month = jan,
year = "2011",
bibdate = "Sat Apr 23 09:24:34 2011",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
note = "Where is this document?",
URL = "http://adegoke.atwebpages.com/",
acknowledgement = ack-nhfb,
}
@Article{Adegoke:2011:NAD,
author = "Kunle Adegoke",
title = "A Novel Approach to the Discovery of Ternary
{BBP}-type Formulas for Polylogarithm Constants",
journal = "Notes on Number Theory and Discrete Mathematics",
volume = "17",
number = "1",
pages = "??--??",
month = "????",
year = "2011",
CODEN = "????",
ISSN = "????",
bibdate = "Sat Apr 23 09:19:08 2011",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
URL = "http://adegoke.atwebpages.com/",
acknowledgement = ack-nhfb,
fjournal = "Notes on Number Theory and Discrete Mathematics",
}
@Unpublished{Adegoke:2011:NDB,
author = "Kunle Adegoke",
title = "New Degree 4 Binary {BBP}-type Formulas and a Zero
Relation",
month = jan,
year = "2011",
bibdate = "Sat Apr 23 09:24:34 2011",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
note = "Where is this document?",
URL = "http://adegoke.atwebpages.com/",
acknowledgement = ack-nhfb,
}
@Article{Adegoke:2011:SRB,
author = "Kunle Adegoke",
title = "Symbolic Routes to {BBP}-type Formulas of any Degree
in Arbitrary Bases",
journal = "Applied Mathematics and Information Sciences",
volume = "??",
number = "??",
pages = "??--??",
month = may,
year = "2011",
bibdate = "Sat Apr 23 09:20:11 2011",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
acknowledgement = ack-nhfb,
fjournal = "Applied Mathematics and Information Sciences",
}
@Article{Almkvist:2011:RLF,
author = "Gert Almkvist",
title = "{Ramanujan}-like formulas for $ 1 / \pi^2 $ and String
Theory [abstract only]",
journal = j-ACM-COMM-COMP-ALGEBRA,
volume = "45",
number = "2",
pages = "92--92",
month = jun,
year = "2011",
CODEN = "????",
DOI = "https://doi.org/10.1145/2016567.2016576",
ISSN = "1932-2232 (print), 1932-2240 (electronic)",
ISSN-L = "1932-2232",
bibdate = "Thu Sep 01 12:20:20 2011",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
note = "To appear in Proceedings of WWCA 2011.",
acknowledgement = ack-nhfb,
fjournal = "ACM Communications in Computer Algebra",
issue = "176",
remark = "The new formula can be used to compute an arbitrary
{\em decimal digit\/} of $ 1 / \pi^2 $ without
computing earlier digits.",
}
@TechReport{Bailey:2011:BTF,
author = "David H. Bailey",
title = "A Compendium of {BBP}-Type Formulas for Mathematical
Constants",
type = "Report",
institution = "Lawrence Berkeley National Laboratory",
address = "Berkeley, CA, USA",
pages = "36",
day = "13",
month = feb,
year = "2011",
bibdate = "Sat Apr 23 09:03:06 2011",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
URL = "http://crd.lbl.gov/~dhbailey/dhbpapers/bbp-formulas.pdf;
http://www.bbp.carma.newcastle.edu.au",
abstract = "A 1996 paper by the author, Peter Borwein and Simon
Plouffe showed that any mathematical constant given by
an infinite series of a certain type has the property
that its $n$-th digit in a particular number base could
be calculated directly, without needing to compute any
of the first $ n - 1$ digits, by means of a simple
algorithm that does not require multiple-precision
arithmetic. Several such formulas were presented in
that paper, including formulas for the constants $ \pi
$ and $ \log 2$. Since then, numerous other formulas of
this type have been found. This paper presents a
compendium of currently known results of this sort,
both proven and conjectured. Experimentally obtained
results which are not yet proven have been checked to
high precision and are marked with a $ \stackrel {?}{ =
}$. Fully established results are as indicated in the
citations and references below.",
acknowledgement = ack-nhfb,
ORCID-numbers = "Bailey, David H./0000-0002-7574-8342",
}
@TechReport{Bailey:2011:CPI,
author = "David H. Bailey and Jonathan M. Borwein and Andrew
Mattingly and Glenn Wightwick",
title = "The Computation of Previously Inaccessible Digits of $
\pi^2 $ and {Catalan's} Constant",
type = "Report",
institution = "Lawrence Berkeley National Laboratory; Centre for
Computer Assisted Research Mathematics and its
Applications (CARMA), University of Newcastle; IBM
Australia",
address = "Berkeley, CA, USA; Callaghan, NSW 2308, Australia; St.
Leonards, NSW 2065, Australia; Pyrmont, NSW 2009,
Australia",
pages = "18",
day = "11",
month = apr,
year = "2011",
bibdate = "Sat Apr 23 08:58:45 2011",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
URL = "http://crd.lbl.gov/~dhbailey/dhbpapers/bbp-bluegene.pdf",
acknowledgement = ack-nhfb,
author-dates = "Jonathan Michael Borwein (20 May 1951--2 August
2016)",
ORCID-numbers = "Bailey, David H./0000-0002-7574-8342; Borwein,
Jonathan/0000-0002-1263-0646",
remark = "Submitted to Notices of the AMS.",
}
@TechReport{Borwein:2011:PSE,
author = "D. Borwein and Jonathan M. Borwein",
title = "Proof of some experimentally conjectured formulas for
$ \pi $",
type = "Preprint",
institution = "Department of Mathematics, University of Western
Ontario and Centre for Computer-assisted Research
Mathematics and its Applications (CARMA), School of
Mathematical and Physical Sciences, University of
Newcastle",
address = "London, ON, Canada and Callaghan, NSW 2308,
Australia",
day = "4",
month = dec,
year = "2011",
bibdate = "Sun Dec 04 10:39:23 2011",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
abstract = "A recent paper by M. Jauregui and C. Tsallis
\cite{Jauregui:2010:NRD}, which explores applications
of the $q$-exponential function and formal
representations of the Dirac function, contains a set
of experimentally discovered formulae for $ \pi $ as
finite series of gamma function ratios. Herein, we
prove rigorously these identities as special cases of
Pfaff--Saalsch{\"u}tz evaluation for $_3 F_2 ({a, b, c}
\atop {d, e} | 1)$ functions. We likewise prove and
extend a corresponding integral identity given in
\cite{Jauregui:2010:NRD}.",
acknowledgement = ack-nhfb,
author-dates = "Jonathan Michael Borwein (20 May 1951--2 August
2016)",
ORCID-numbers = "Borwein, Jonathan/0000-0002-1263-0646",
}
@Article{Chu:2011:DBS,
author = "Wenchang Chu",
title = "{Dougall}'s bilateral {$_2 H_2$-series} and
{Ramanujan}-like $ \pi $-formulae",
journal = j-MATH-COMPUT,
volume = "80",
number = "276",
pages = "2223--2251",
month = oct,
year = "2011",
CODEN = "MCMPAF",
ISSN = "0025-5718 (print), 1088-6842 (electronic)",
ISSN-L = "0025-5718",
bibdate = "Mon Oct 24 10:33:34 MDT 2011",
bibsource = "http://www.ams.org/mcom/2011-80-276;
https://www.math.utah.edu/pub/tex/bib/mathcomp2010.bib;
https://www.math.utah.edu/pub/tex/bib/pi.bib",
URL = "http://www.ams.org/journals/mcom/2011-80-276/S0025-5718-2011-02474-9/home.html;
http://www.ams.org/journals/mcom/2011-80-276/S0025-5718-2011-02474-9/S0025-5718-2011-02474-9.pdf;
http://www.ams.org/mathscinet-getitem?mr=2813357",
abstract = "The modified Abel lemma on summation by parts is
employed to investigate the partial sum of Dougall's
bilateral $_2 H_2$-series. Several unusual
transformations into fast convergent series are
established. They lead surprisingly to numerous
infinite series expressions for $ \pi $, including
several formulae discovered by Ramanujan (1914) and
recently by Guillera (2008).",
acknowledgement = ack-nhfb,
fjournal = "Mathematics of Computation",
journal-URL = "http://www.ams.org/mcom/",
}
@Misc{Knuth:2011:WPC,
author = "Donald Knuth",
title = "Why Pi? [{Christmas} tree lecture]",
howpublished = "100-minute YouTube video.",
day = "6",
month = sep,
year = "2011",
bibdate = "Fri Sep 30 15:50:21 2016",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
URL = "https://www.youtube.com/watch?v=mw6dYK9LRzU",
acknowledgement = ack-nhfb,
}
@Unpublished{Lafont:2011:DBT,
author = "Jaume Oliver Lafont",
title = "Degree $1$ {BBP}-Type Zero Relations",
day = "27",
month = jan,
year = "2011",
bibdate = "Sat Apr 23 09:16:32 2011",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
note = "Where is this document?",
acknowledgement = ack-nhfb,
}
@Unpublished{Yee:2011:LC,
author = "Alexander Yee",
title = "Large Computations",
day = "7",
month = mar,
year = "2011",
bibdate = "Sat Apr 23 10:04:00 2011",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
note = "Where is this document?",
URL = "http://www.numberworld.org/nagisa_runs/computations.html",
acknowledgement = ack-nhfb,
}
@Unpublished{Yee:2011:TDPa,
author = "Alexander Yee and Shigeru Kondo",
title = "Trillion Digits of Pi --- New World Record",
day = "7",
month = mar,
year = "2011",
bibdate = "Sat Apr 23 10:04:53 2011",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
note = "Where is this document?",
URL = "http://www.numberworld.org/misc_runs/pi-5t/details.html",
acknowledgement = ack-nhfb,
}
@TechReport{Yee:2011:TDPb,
author = "Alexander J. Yee and Shigeru Kondo",
title = "10 Trillion Digits of Pi: A Case Study of Summing
Hypergeometric Series to High Precision on Multicore
Systems",
type = "Preprint",
institution = "University of Illinois Urbana-Champaign and Asahimatsu
Food Co. Ltd.",
address = "Urbana, IL, USA and Iida, Japan",
year = "2011",
bibdate = "Fri May 03 18:47:53 2013",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
URL = "http://hdl.handle.net/2142/28348",
abstract = "Hypergeometric series are powerful mathematical tools
with many usages. Many mathematical functions, such as
trigonometric functions, can be partly or entirely
expressed in terms of them. In most cases this allows
efficient evaluation of such functions, their
derivatives and their integrals. They are also the most
efficient way known to compute constants, such as $ \pi
$ and $e$, to high precision. Binary splitting is a low
complexity algorithm for summing up hypergeometric
series. It is a divide-and-conquer algorithm and can
therefore be parallelized. However, it requires large
number arithmetic, increases memory usage, and exhibits
asymmetric workload, which makes it non-trivial to
parallelize. We describe a high performing parallel
implementation of the binary splitting algorithm for
summing hypergeometric series on shared-memory
multicores. To evaluate the implementation we have
computed $ \pi $ to 5 trillion digits in August 2010
and 10 trillion digits in October 2011 both of which
were new world records. Furthermore, the implementation
techniques described in this paper are general, and can
be used to implement applications in other domains that
exhibit similar features.",
acknowledgement = ack-nhfb,
}
@Article{Zorzi:2011:BLP,
author = "Alberto Zorzi",
title = "{Benford's law} and pi",
journal = j-MATH-GAZ,
volume = "95",
number = "533",
pages = "264--266",
month = jul,
year = "2011",
CODEN = "MAGAAS",
DOI = "????",
ISSN = "0025-5572",
bibdate = "Mon Feb 18 18:59:42 2013",
bibsource = "https://www.math.utah.edu/pub/tex/bib/benfords-law.bib;
https://www.math.utah.edu/pub/tex/bib/pi.bib",
URL = "????",
acknowledgement = ack-nhfb,
fjournal = "Mathematical Gazette",
journal-URL = "http://www.m-a.org.uk/jsp/index.jsp?lnk=620",
remark = "The journal Web site lacks a search feature, and the
archives only cover up to 2007. JSTOR has only issues
up to 2007.",
}
@Article{Amdeberhan:2012:FEC,
author = "Tewodros Amdeberhan and David Borwein and Jonathan M.
Borwein and Armin Straub",
title = "On formulas for $ \pi $ experimentally conjectured by
{Jauregui--Tsallis}",
journal = j-J-MATH-PHYS,
volume = "53",
number = "7",
pages = "073708",
month = jul,
year = "2012",
CODEN = "JMAPAQ",
DOI = "https://doi.org/10.1063/1.4735283",
ISSN = "0022-2488 (print), 1089-7658 (electronic), 1527-2427",
ISSN-L = "0022-2488",
bibdate = "Thu Nov 8 12:34:42 MST 2012",
bibsource = "http://jmp.aip.org/;
https://www.math.utah.edu/pub/tex/bib/jmathphys2010.bib;
https://www.math.utah.edu/pub/tex/bib/pi.bib",
URL = "http://jmp.aip.org/resource/1/jmapaq/v53/i7/p073708_s1",
acknowledgement = ack-nhfb,
author-dates = "Jonathan Michael Borwein (20 May 1951--2 August
2016)",
fjournal = "Journal of Mathematical Physics",
journal-URL = "http://jmp.aip.org/",
onlinedate = "18 July 2012",
ORCID-numbers = "Borwein, Jonathan/0000-0002-1263-0646",
}
@Article{Bailey:2012:EAN,
author = "David H. Bailey and Jonathan M. Borwein and Cristian
S. Calude and Michael J. Dinneen and Monica Dumitrescu
and Alex Yee",
title = "An Empirical Approach to the Normality of $ \pi $",
journal = j-EXP-MATH,
volume = "21",
number = "4",
pages = "375--384",
year = "2012",
DOI = "https://doi.org/10.1080/10586458.2012.665333",
ISSN = "1058-6458 (print), 1944-950X (electronic)",
ISSN-L = "1058-6458",
bibdate = "Thu May 2 18:39:41 MDT 2013",
bibsource = "https://www.math.utah.edu/pub/tex/bib/expmath.bib;
https://www.math.utah.edu/pub/tex/bib/pi.bib",
acknowledgement = ack-nhfb,
author-dates = "Jonathan Michael Borwein (20 May 1951--2 August
2016)",
fjournal = "Experimental Mathematics",
journal-URL = "http://www.tandfonline.com/loi/uexm20",
ORCID-numbers = "Bailey, David H./0000-0002-7574-8342; Borwein,
Jonathan/0000-0002-1263-0646",
}
@Article{Bailey:2012:HPC,
author = "D. H. Bailey and R. Barrio and J. M. Borwein",
title = "High-precision computation: {Mathematical} physics and
dynamics",
journal = j-APPL-MATH-COMP,
volume = "218",
number = "20",
pages = "10106--10121",
day = "15",
month = jun,
year = "2012",
CODEN = "AMHCBQ",
DOI = "https://doi.org/10.1016/j.amc.2012.03.087",
ISSN = "0096-3003 (print), 1873-5649 (electronic)",
ISSN-L = "0096-3003",
bibdate = "Mon May 14 07:47:47 MDT 2012",
bibsource = "https://www.math.utah.edu/pub/tex/bib/applmathcomput2010.bib;
https://www.math.utah.edu/pub/tex/bib/pi.bib;
http://www.sciencedirect.com/science/journal/00963003",
URL = "http://www.sciencedirect.com/science/article/pii/S0096300312003505",
acknowledgement = ack-nhfb,
author-dates = "Jonathan Michael Borwein (20 May 1951--2 August
2016)",
fjournal = "Applied Mathematics and Computation",
journal-URL = "http://www.sciencedirect.com/science/journal/00963003",
ORCID-numbers = "Bailey, David H./0000-0002-7574-8342; Borwein,
Jonathan/0000-0002-1263-0646",
}
@Article{Fuks:2012:AAK,
author = "Henryk Fuk{\'s}",
title = "{Adam Adamandy Kocha{\'n}ski}'s Approximations of $
\pi $: Reconstruction of the Algorithm",
journal = j-MATH-INTEL,
volume = "34",
number = "4",
pages = "40--45",
month = "????",
year = "2012",
CODEN = "MAINDC",
DOI = "https://doi.org/10.1007/s00283-012-9312-1",
ISSN = "0343-6993 (print), 1866-7414 (electronic)",
ISSN-L = "0343-6993",
bibdate = "Thu Feb 14 06:21:44 2013",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
URL = "http://arxiv.org/abs/1111.1739;
http://link.springer.com/article/10.1007/s00283-012-9312-1",
acknowledgement = ack-nhfb,
fjournal = "The Mathematical Intelligencer",
keywords = "Adam Adamandy Kocha{\'n}ski, S.J. (1631--1700); Online
Encyclopedia of Integer Sequence A191642",
remark = "The author examines Kocha{\'n}ski's investigations of
the calculation of $ \pi $ by successive integer
approximations, and shows that had Kocha{\'n}ski made a
minor change in one of his generator sequences, he
would have discovered convergents and continued
fractions several decades before they were published by
John Wallis in his 1695 book, \booktitle{Opera
Mathematica}. Kocha{\'n}ski's unpublished papers were
held by the National Library in Warsaw, and lost in
1944 when it was set on fire by Nazi occupiers during
the Warsaw Uprising.",
}
@Article{Osada:2012:EHC,
author = "Naoki Osada",
title = "The early history of convergence acceleration
methods",
journal = j-NUMER-ALGORITHMS,
volume = "60",
number = "2",
pages = "205--221",
month = jun,
year = "2012",
CODEN = "NUALEG",
DOI = "https://doi.org/10.1007/s11075-012-9539-0",
ISSN = "1017-1398 (print), 1572-9265 (electronic)",
ISSN-L = "1017-1398",
bibdate = "Wed Mar 6 09:09:43 MST 2013",
bibsource = "http://springerlink.metapress.com/openurl.asp?genre=issue&issn=1017-1398&volume=60&issue=2;
https://www.math.utah.edu/pub/tex/bib/numana2010.bib;
https://www.math.utah.edu/pub/tex/bib/numeralgorithms.bib;
https://www.math.utah.edu/pub/tex/bib/pi.bib",
URL = "http://www.springerlink.com/openurl.asp?genre=article&issn=1017-1398&volume=60&issue=2&spage=205",
acknowledgement = ack-nhfb,
fjournal = "Numerical Algorithms",
journal-URL = "http://link.springer.com/journal/11075",
keywords = "A. C. Aitken (1895--1967); Aitken Delta-squared
process; Archimedes (287BCE--212BCE); Christiaan
Huygens (1629--1695); convergence acceleration;
Cyclometricus (1621); De ciculi magnitudine inventa
(1654); history of numerical analysis; Isaac Newton;
Katahiro Takebe; Ludolf van Ceulen (1540--1610); pi
calculation; Richardson extrapolation; sequence of
intervals; Shigekiyo Muramatsu; Suanxue Qimeng
(Mathematical Enlightenment) (1299); Takakazu Seki
(????--1708); Willebrord Snell (1580--1626); Yosimasu
Murase; Zhu Shijie",
remark-1 = "This paper gives a nice historical survey of work in
Japan in the 1600s and 1700s on methods for computing
$\pi$, and the volume of a sphere, which led to the
discovery of extrapolation procedures that were later
independently rediscovered in Europe, and credited to
European scientists. It is unclear from the article
whether those early Japanese discoveries influenced
later work in Japan, or were lost until historians
found them in the late Twentieth Century.",
remark-2 = "From page 214: ``The Aitken $\Delta^2$ process was
discovered by Japanese mathematician Takakazu Seke
(?--1708) before 1680.''.",
remark-3 = "From pages 214--215: ``The first Japanese
mathematician who determined the circumference ratio
was Shigekiyo Muramatsu. In 1663 he computed \ldots{}
$\pi \approx 3.14159\,264\ldots{}''.",
remark-4 = "From page 215: ``In 1673 Yosimasu Murase determined
$\pi$ as 3.1415.''",
remark-5 = "From page 217: ``[In 1712, Takakazu] Seki derived the
rational approximate $355 / 113 (\approx 3.141592)$ of
$\pi$.",
remark-6 = "From pages 218 and 220: ``The Richardson extrapolation
process was discovered by [Takakazu] Seki's disciple
Katahiro Takebe before 1710, probably before 1695.''",
remark-7 = "From page 220: In 1720, Katahiro Takebe found $\pi =
3.14159\,26535\,89793\,23846\,2643. ``[Katahiro] Takebe
gave exact 41 decimal digits [of $\pi$].''",
}
@Article{Shelburne:2012:ED,
author = "Brian J. Shelburne",
title = "The {ENIAC}'s 1949 Determination of $ \pi $",
journal = j-IEEE-ANN-HIST-COMPUT,
volume = "34",
number = "3",
pages = "44--54",
month = jul # "\slash " # sep,
year = "2012",
CODEN = "IAHCEX",
DOI = "https://doi.org/10.1109/MAHC.2011.61",
ISSN = "1058-6180 (print), 1934-1547 (electronic)",
ISSN-L = "1058-6180",
bibdate = "Mon Oct 22 07:04:43 2012",
bibsource = "https://www.math.utah.edu/pub/tex/bib/ieeeannhistcomput.bib;
https://www.math.utah.edu/pub/tex/bib/pi.bib",
abstract = "In January 1950, George W. Reitwiesner published ``An
ENIAC Determination of $ \pi $ and $e$ to more than
2000 Decimal Places'' in Mathematical Tables and Other
Aides to Computation \cite{Reitwiesner:1950:EDM} which
described the first use of a computer, the ENIAC, to
calculate the decimal expansion of $ \pi $. Since the
history of $ \pi $ stretches back over thousands of
years, the use of the ENIAC to determine $ \pi $ is an
important historical and technological milestone. It is
especially interesting since the ENIAC was not designed
to perform this type of calculation as it could only
store 200 decimal digits while the determination of e
and $ \pi $ required manipulating numbers 2000+ digits
long. Starting with Reitwiesner's description of the
calculation, the known architecture of the ENIAC, how
it was programmed, and the mathematics used, we examine
why the calculation was undertaken, how the calculation
had to be done, and what was subsequently learned.",
acknowledgement = ack-nhfb,
fjournal = "IEEE Annals of the History of Computing",
journal-URL = "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=85",
pdfdate = "8 August 2011",
remark = "This paper contains an interesting survey of work on
the calculation of $ \pi $ up to the early 1950s, with
a detailed reconstruction of its determination on the
ENIAC. From page 1 of the paper: ``Early in June, 1949,
Professor John von Neumann expressed an interest in the
possibility that the ENIAC might sometime be employed
to determine the value of $ \pi $ and $e$ to many
decimal places with a view toward obtaining a
statistical measure of the randomness of the
distribution of the digits.'' From page 2: ``\ldots{}
Augustus De Morgan (1806--1871) who noticed the smaller
number of appearances of the digit 7 in Shank's 607
digit determination of $ \pi $. It was later determined
that Shank's determination had an error beginning at
the 528th digit.'' From page 11: ``A preliminary
investigation has indicated that the digits of $e$
deviate significantly from randomness (in the sense of
staying closer to their expected values than a random
sequence of this length normally would) while for $ \pi
$ no significant deviations have so far been
detected.'' See \cite{Metropolis:1950:STV} for that
analysis.",
}
@Article{Agarwal:2013:BGC,
author = "Ravi P. Agarwal and Hans Agarwal and Syamal K. Sen",
title = "Birth, growth and computation of pi to ten trillion
digits",
journal = j-ADV-DIFFERENCE-EQU,
volume = "2013",
number = "100",
pages = "1--59",
year = "2013",
CODEN = "????",
DOI = "https://doi.org/10.1186/1687-1847-2013-100",
ISSN = "1687-1847",
ISSN-L = "1687-1847",
bibdate = "Mon Jan 06 10:25:51 2014",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
URL = "http://www.advancesindifferenceequations.com/content/2013/1/100",
acknowledgement = ack-nhfb,
fjournal = "Advances in Difference Equations",
journal-URL = "http://www.advancesindifferenceequations.com/",
}
@InCollection{Alladi:2013:R,
author = "Krishnaswami Alladi",
title = "{Ramanujan} and $ \pi $",
crossref = "Alladi:2013:RPW",
pages = "103--109",
year = "2013",
DOI = "https://doi.org/10.1007/978-81-322-0767-2_16",
bibdate = "Sat Sep 03 17:04:24 2016",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
note = "This article appeared in \booktitle{The Hindu}, India
s national newspaper in December 1994 on Ramanujan's
107th birth anniversary.",
URL = "http://link.springer.com/chapter/10.1007/978-81-322-0767-2_16",
acknowledgement = ack-nhfb,
}
@Article{AragonArtacho:2013:WRN,
author = "Francisco {Arag{\'o}n Artacho} and David H. Bailey and
Jonathan M. Borwein and Peter B. Borwein",
title = "Walking on Real Numbers",
journal = j-MATH-INTEL,
volume = "35",
number = "1",
pages = "42--60",
month = mar,
year = "2013",
CODEN = "MAINDC",
DOI = "https://doi.org/10.1007/s00283-012-9340-x",
ISSN = "0343-6993 (print), 1866-7414 (electronic)",
ISSN-L = "0343-6993",
bibdate = "Fri Mar 15 11:52:16 2013",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib;
https://www.math.utah.edu/pub/tex/bib/prng.bib",
URL = "http://gigapan.com/gigapans/106803;
http://www.davidhbailey.com/dhbpapers/tools-walk.pdf",
acknowledgement = ack-nhfb,
author-dates = "Jonathan Michael Borwein (20 May 1951--2 August
2016)",
fjournal = "The Mathematical Intelligencer",
keywords = "Catalan's constant; Champernowne numbers; continued
fractions; Copeland--Erd{\H{o}}s numbers; DNA genome
numbers; dragon curves; Erd{\H{o}}s--Borwein numbers;
Euler--Mascherino constant ($\gamma$); expected
random-walk distance; exponential constant ($e$);
Fibonacci constant ($F$); Gauss--Kuzmin distribution;
irrational numbers; Koch snowflakes; Liouville number
($\lambda_2$); logarithmic constant ($\log 2$);
Minkowski--Bouligand dimension; normal numbers;
normalized random-walk distance; paper-folding
constant; paper-folding numbers; pi (number); random
walks; Riemann zeta numbers ($\zeta(n)$);
self-similarity; Stoneham numbers; strong normality;
Thue--Morse numbers; transcendental numbers; turtle
plots",
ORCID-numbers = "Bailey, David H./0000-0002-7574-8342; Borwein,
Jonathan/0000-0002-1263-0646",
}
@Misc{Bailey:2013:CBT,
author = "David H. Bailey",
title = "A compendium of {BBP}-type formulas for mathematical
constants",
howpublished = "Web site.",
day = "29",
month = apr,
year = "2013",
bibdate = "Tue Apr 21 16:16:46 2020",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
URL = "http://www.davidhbailey.com/dhbpapers/bbp-formulas.pdf",
acknowledgement = ack-nhfb,
}
@Article{Bailey:2013:CPI,
author = "David H. Bailey and Jonathan M. Borwein and Andrew
Mattingly and Glenn Wightwick",
title = "The computation of previously inaccessible digits of
$ \pi^2 $ and {Catalan}'s constant",
journal = j-NAMS,
volume = "60",
number = "7",
pages = "844--854",
month = aug,
year = "2013",
CODEN = "AMNOAN",
DOI = "https://doi.org/10.1090/noti1015",
ISSN = "0002-9920 (print), 1088-9477 (electronic)",
ISSN-L = "0002-9920",
MRclass = "11Y60 (65-04)",
MRnumber = "3086394",
MRreviewer = "Michael M. Dediu",
bibdate = "Wed Aug 10 11:09:47 2016",
bibsource = "https://www.math.utah.edu/pub/bibnet/authors/b/borwein-jonathan-m.bib;
https://www.math.utah.edu/pub/tex/bib/pi.bib",
URL = "http://docserver.carma.newcastle.edu.au/1436/;
http://www.ams.org/notices/201307/rnoti-p844.pdf",
acknowledgement = ack-nhfb,
author-dates = "Jonathan Michael Borwein (20 May 1951--2 August
2016)",
fjournal = "Notices of the American Mathematical Society",
journal-URL = "http://www.ams.org/notices/",
ORCID-numbers = "Bailey, David H./0000-0002-7574-8342; Borwein,
Jonathan/0000-0002-1263-0646",
}
@Article{Bailey:2013:DPR,
author = "David H. Bailey and Jonathan M. Borwein",
title = "Are the Digits of Pi Random?",
journal = "Huffington Post",
day = "16",
month = apr,
year = "2013",
bibdate = "Wed Apr 17 08:22:02 2013",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
URL = "http://www.huffingtonpost.com/david-h-bailey/are-the-digits-of-pi-random_b_3085725.html",
acknowledgement = ack-nhfb,
author-dates = "Jonathan Michael Borwein (20 May 1951--2 August
2016)",
ORCID-numbers = "Bailey, David H./0000-0002-7574-8342; Borwein,
Jonathan/0000-0002-1263-0646",
}
@TechReport{Bailey:2013:PDU,
author = "David H. Bailey and Jonathan Borwein",
title = "Pi Day is upon us again and we still do not know if Pi
is normal",
type = "Report",
institution = "Lawrence Berkeley National Laboratory and Centre for
Computer Assisted Research Mathematics and its
Applications (CARMA), University of Newcastle",
address = "Berkeley, CA 94720, USA and Callaghan, NSW 2308,
Australia",
pages = "20",
day = "29",
month = may,
year = "2013",
bibdate = "Mon Jun 10 07:23:57 2013",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
URL = "http://www.carma.newcastle.edu.au/jon/pi-monthly.pdf",
acknowledgement = ack-nhfb,
author-dates = "Jonathan Michael Borwein (20 May 1951--2 August
2016)",
ORCID-numbers = "Bailey, David H./0000-0002-7574-8342; Borwein,
Jonathan/0000-0002-1263-0646",
}
@Article{Beliakov:2013:EIBa,
author = "Gleb Beliakov and Michael Johnstone and Doug Creighton
and Tim Wilkin",
title = "An efficient implementation of {Bailey} and
{Borwein}'s algorithm for parallel random number
generation on graphics processing units",
journal = j-COMPUTING,
volume = "95",
number = "4",
pages = "309--326",
month = apr,
year = "2013",
CODEN = "CMPTA2",
DOI = "https://doi.org/10.1007/s00607-012-0234-8",
ISSN = "0010-485X (print), 1436-5057 (electronic)",
ISSN-L = "0010-485X",
bibdate = "Tue May 7 12:18:19 MDT 2013",
bibsource = "http://springerlink.metapress.com/openurl.asp?genre=issue&issn=0010-485X&volume=95&issue=4;
https://www.math.utah.edu/pub/tex/bib/compphyscomm2010.bib;
https://www.math.utah.edu/pub/tex/bib/computing.bib;
https://www.math.utah.edu/pub/tex/bib/pi.bib;
https://www.math.utah.edu/pub/tex/bib/prng.bib",
note = "See also \cite{Beliakov:2013:EIBb}.",
URL = "http://link.springer.com/article/10.1007/s00607-012-0234-8",
acknowledgement = ack-nhfb,
fjournal = "Computing",
journal-URL = "http://link.springer.com/journal/607",
keywords = "$\alpha_{2,3}$; normal number",
subject-dates = "Jonathan Michael Borwein (20 May 1951--2 August
2016)",
}
@Article{Beliakov:2013:EIBb,
author = "G. Beliakov and D. Creighton and M. Johnstone and T.
Wilkin",
title = "Efficient implementation of {Bailey} and {Borwein}
pseudo-random number generator based on normal
numbers",
journal = j-COMP-PHYS-COMM,
volume = "184",
number = "8",
pages = "1999--2004",
month = aug,
year = "2013",
CODEN = "CPHCBZ",
DOI = "https://doi.org/10.1016/j.cpc.2013.03.019",
ISSN = "0010-4655 (print), 1879-2944 (electronic)",
ISSN-L = "0010-4655",
bibdate = "Wed May 15 07:02:08 MDT 2013",
bibsource = "https://www.math.utah.edu/pub/tex/bib/compphyscomm2010.bib;
https://www.math.utah.edu/pub/tex/bib/pi.bib;
https://www.math.utah.edu/pub/tex/bib/prng.bib",
note = "See also \cite{Beliakov:2013:EIBa}.",
URL = "http://www.sciencedirect.com/science/article/pii/S0010465513001276",
acknowledgement = ack-nhfb,
fjournal = "Computer Physics Communications",
journal-URL = "http://www.sciencedirect.com/science/journal/00104655",
subject-dates = "Jonathan Michael Borwein (20 May 1951--2 August
2016)",
}
@Article{Casey:2013:PPP,
author = "Stephen D. Casey and Brian M. Sadler",
title = "Pi, the Primes, Periodicities, and Probability",
journal = j-AMER-MATH-MONTHLY,
volume = "120",
number = "7",
pages = "594--608",
month = aug,
year = "2013",
CODEN = "AMMYAE",
DOI = "https://doi.org/10.4169/amer.math.monthly.120.07.594",
ISSN = "0002-9890 (print), 1930-0972 (electronic)",
ISSN-L = "0002-9890",
bibdate = "Tue Mar 4 06:16:39 MST 2014",
bibsource = "http://www.jstor.org/journals/00029890.html;
http://www.jstor.org/stable/10.4169/amermathmont.120.issue-07;
https://www.math.utah.edu/pub/tex/bib/amermathmonthly2010.bib;
https://www.math.utah.edu/pub/tex/bib/pi.bib",
URL = "http://www.jstor.org/stable/pdfplus/10.4169/amer.math.monthly.120.07.594.pdf",
acknowledgement = ack-nhfb,
fjournal = "American Mathematical Monthly",
journal-URL = "https://www.jstor.org/journals/00029890.htm",
}
@Misc{Gourevitch:2013:W,
author = "Boris Gour{\'e}vitch",
title = "The world of $ \pi $",
howpublished = "Web site",
day = "13",
month = apr,
year = "2013",
bibdate = "Mon Jun 15 07:53:57 2020",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
URL = "http://www.pi314.net/eng/index.php",
acknowledgement = ack-nhfb,
remark = "Also available in original French version,
\booktitle{L'univers de $ \pi $}.",
}
@Misc{Karrels:2013:CDC,
author = "Ed Karrels",
title = "Computing digits of $ \pi $ with {CUDA}",
type = "Web site.",
year = "2013",
bibdate = "Mon Jun 10 08:24:23 2013",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
URL = "http://www.karrels.org/pi",
acknowledgement = ack-nhfb,
remark = "From the introduction: 2013-05-23 Four Quadrillionth
and counting\ldots{}: After 32 days and 35,000 hours of
GPU time (and another 32 days and 35,000 hours to
doublecheck), my computation of the four quadrillionth
digit of $ \pi $ has finished. Starting at the four
quadrillionth hexadecimal digit of $ \ii $, the next
eight digits are {\tt 5cc37dec}.",
}
@InProceedings{Karrels:2013:SCQ,
author = "Ed Karrels",
editor = "????",
booktitle = "{GPU Technology Conference, March 18--21, 2013, San
Jose, California}",
title = "S3071 --- Computing the Quadrillionth Digit of Pi: A
Supercomputer in the Garage",
publisher = "????",
address = "????",
year = "2013",
bibdate = "Mon Jun 10 08:28:36 2013",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
URL = "http://registration.gputechconf.com/quicklink/2IXnrGH",
abstract = "In 1995, Bailey, Borwein and Plouffe discovered a new
formula for computing pi that ignited a computation
arms race by making it possible to compute digits of pi
without storing previous digits, and without the use of
large-number arithmetic. In 2010 Yahoo! set a world
record, using a variant of the Bailey--Borwein--Plouffe
formula on an 8000-core Hadoop cluster to compute the
two quadrillionth bit of pi. In this talk, I'll discuss
how I stole the record from Yahoo! by computing the
four quadrillionth bit of pi on a single CUDA-enabled
computer.",
acknowledgement = ack-nhfb,
}
@Article{Ritelli:2013:API,
author = "Daniele Ritelli",
title = "Another Proof of $ {\zeta (2) = \frac {\pi^2}{6}} $
Using Double Integrals",
journal = j-AMER-MATH-MONTHLY,
volume = "120",
number = "7",
pages = "642--645",
month = aug,
year = "2013",
CODEN = "AMMYAE",
DOI = "https://doi.org/10.4169/amer.math.monthly.120.07.642",
ISSN = "0002-9890 (print), 1930-0972 (electronic)",
ISSN-L = "0002-9890",
bibdate = "Tue Mar 4 06:16:39 MST 2014",
bibsource = "http://www.jstor.org/journals/00029890.html;
http://www.jstor.org/stable/10.4169/amermathmont.120.issue-07;
https://www.math.utah.edu/pub/tex/bib/amermathmonthly2010.bib;
https://www.math.utah.edu/pub/tex/bib/pi.bib",
URL = "http://www.jstor.org/stable/pdfplus/10.4169/amer.math.monthly.120.07.642.pdf",
acknowledgement = ack-nhfb,
fjournal = "American Mathematical Monthly",
journal-URL = "https://www.jstor.org/journals/00029890.htm",
}
@Article{Wan:2013:HGF,
author = "James G. Wan",
title = "Hypergeometric generating functions and series for $ 1
/ \pi $",
journal = j-ACM-COMM-COMP-ALGEBRA,
volume = "47",
number = "3--4",
pages = "114--115",
month = sep,
year = "2013",
CODEN = "????",
DOI = "https://doi.org/10.1145/2576802.2576820",
ISSN = "1932-2232 (print), 1932-2240 (electronic)",
ISSN-L = "1932-2232",
bibdate = "Tue Jan 28 17:13:26 MST 2014",
bibsource = "http://portal.acm.org/;
https://www.math.utah.edu/pub/tex/bib/pi.bib;
https://www.math.utah.edu/pub/tex/bib/sigsam.bib",
acknowledgement = ack-nhfb,
fjournal = "ACM Communications in Computer Algebra",
}
@Misc{Yee:2013:IST,
author = "Alexander Yee and Shiguro Kondo",
title = "It Stands at 10 trillion digits of Pi\ldots{} World
Record for both Desktop and Supercomputer!!!",
howpublished = "Web site",
day = "15",
month = apr,
year = "2013",
bibdate = "Wed Apr 17 08:27:32 2013",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
note = "This site also contains a table of digit records from
2009 to 2013 for various mathematical constants. The $
\pi $ record of 10,000,000,000,050 decimal digits was
reached on 17 October 2011 after 371 days of
computation, and 45 hours of verification.",
URL = "http://www.numberworld.org/y-cruncher/",
acknowledgement = ack-nhfb,
}
@Article{Bailey:2014:PDU,
author = "David H. Bailey and Jonathan Borwein",
title = "Pi Day Is Upon Us Again and We Still Do Not Know if Pi
Is Normal",
journal = j-AMER-MATH-MONTHLY,
volume = "121",
number = "3",
pages = "191--206",
month = mar,
year = "2014",
CODEN = "AMMYAE",
DOI = "https://doi.org/10.4169/amer.math.monthly.121.03.191",
ISSN = "0002-9890 (print), 1930-0972 (electronic)",
ISSN-L = "0002-9890",
bibdate = "Tue Mar 4 06:16:50 MST 2014",
bibsource = "http://www.jstor.org/journals/00029890.html;
http://www.jstor.org/stable/10.4169/amermathmont.121.issue-03;
https://www.math.utah.edu/pub/tex/bib/amermathmonthly2010.bib;
https://www.math.utah.edu/pub/tex/bib/pi.bib",
URL = "http://www.jstor.org/stable/pdfplus/10.4169/amer.math.monthly.121.03.191.pdf",
acknowledgement = ack-nhfb,
author-dates = "Jonathan Michael Borwein (20 May 1951--2 August
2016)",
fjournal = "American Mathematical Monthly",
journal-URL = "https://www.jstor.org/journals/00029890.htm",
ORCID-numbers = "Bailey, David H./0000-0002-7574-8342; Borwein,
Jonathan/0000-0002-1263-0646",
}
@InCollection{Borwein:2014:LPA,
author = "Jonathan M. Borwein",
title = "The Life of Pi: From {Archimedes} to {ENIAC} and
Beyond",
crossref = "Sidoli:2014:ATB",
pages = "531--561",
year = "2014",
bibdate = "Tue Mar 04 14:32:29 2014",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
acknowledgement = ack-nhfb,
author-dates = "Jonathan Michael Borwein (20 May 1951--2 August
2016)",
ORCID-numbers = "Borwein, Jonathan/0000-0002-1263-0646",
}
@Article{Ganz:2014:DES,
author = "Reinhard E. Ganz",
title = "The Decimal Expansion of $ \pi $ Is Not Statistically
Random",
journal = j-EXP-MATH,
volume = "23",
number = "2",
pages = "99--104",
year = "2014",
CODEN = "????",
DOI = "https://doi.org/10.1080/10586458.2013.870504",
ISSN = "1058-6458 (print), 1944-950X (electronic)",
ISSN-L = "1058-6458",
bibdate = "Wed Sep 10 07:36:52 MDT 2014",
bibsource = "https://www.math.utah.edu/pub/bibnet/authors/b/borwein-jonathan-m.bib;
https://www.math.utah.edu/pub/tex/bib/expmath.bib;
https://www.math.utah.edu/pub/tex/bib/pi.bib;
http://www.tandfonline.com/toc/uexm20/23/2",
note = "See the reproduction of results, and reanalysis, in
\cite{Bailey:2016:RCS}, that reveals a flaw in the
statistical analysis in this paper: Ganz used only a
single blocksize in sampling digits, and that blocksize
produces anomalous statistics.",
acknowledgement = ack-nhfb,
fjournal = "Experimental Mathematics",
journal-URL = "http://www.tandfonline.com/loi/uexm20",
remark-1 = "From page 100, column 2: ``\ldots{} This resulted in a
test of third-order symbolic statistics, which the
decimal expansion of $\pi$ fails at a high level of
statistical significance.''",
remark-2 = "From page 104, column 1: ``Thus, the decimal expansion
of $\pi$ cannot considered the realization of a
sequence of iid random variables with uniform
distribution on $\{0, 1, \ldots{}, 9\}$.''",
remark-3 = "Graphs on page 103 of third-order statistics on the
first $10^{13}$ digits of $\pi$ have quite different
appearance before and after digit shuffling. The
statistics after the shuffle more nearly resemble a
normal distribution, whereas before, they are clearly
skewed.",
}
@Article{Lee:2014:HPD,
author = "Jolie Lee",
title = "Happy Pi Day! {Unless} you are a Tauist",
journal = "USA Today",
day = "17",
month = mar,
year = "2014",
ISSN = "0734-7456",
ISSN-L = "0734-7456",
bibdate = "Tue Mar 18 17:27:55 2014",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
URL = "http://www.usatoday.com/story/news/nation-now/2014/03/14/pi-day-tau-math/6410959/",
acknowledgement = ack-nhfb,
journal-URL = "http://www.usatoday.com/",
keywords = "Bob Palais; Michael Hartl; pi day; tau day",
}
@Article{Papadopoulos:2014:HAH,
author = "Ioannis Papadopoulos",
title = "How {Archimedes} Helped Students to Unravel the
Mystery of the Magical Number Pi",
journal = j-SCI-EDUC-SPRINGER,
volume = "23",
number = "1",
pages = "61--77",
month = jan,
year = "2014",
CODEN = "SCEDE9",
DOI = "https://doi.org/10.1007/s11191-013-9643-0",
ISSN = "0926-7220 (print), 1573-1901 (electronic)",
ISSN-L = "0926-7220",
bibdate = "Mon Jun 19 11:34:31 MDT 2017",
bibsource = "http://link.springer.com/journal/11191/23/1;
https://www.math.utah.edu/pub/tex/bib/pi.bib;
https://www.math.utah.edu/pub/tex/bib/sci-educ-springer.bib",
acknowledgement = ack-nhfb,
fjournal = "Science \& Education (Springer)",
journal-URL = "http://link.springer.com/journal/11191",
}
@Article{Borwein:2015:PPA,
author = "Jonathan Borwein and Scott Chapman",
title = "{I} Prefer Pi: Addenda",
journal = j-AMER-MATH-MONTHLY,
volume = "122",
number = "8",
pages = "800--800",
month = oct,
year = "2015",
CODEN = "AMMYAE",
DOI = "https://doi.org/10.4169/amer.math.monthly.122.8.800",
ISSN = "0002-9890 (print), 1930-0972 (electronic)",
ISSN-L = "0002-9890",
bibdate = "Tue Oct 20 06:20:38 MDT 2015",
bibsource = "https://www.math.utah.edu/pub/tex/bib/amermathmonthly2010.bib;
https://www.math.utah.edu/pub/tex/bib/pi.bib",
note = "See \cite{Borwein:2015:PPB}.",
URL = "http://www.jstor.org/stable/10.4169/amer.math.monthly.122.8.800",
acknowledgement = ack-nhfb,
author-dates = "Jonathan Michael Borwein (20 May 1951--2 August
2016)",
fjournal = "American Mathematical Monthly",
journal-URL = "http://www.jstor.org/journals/00029890.html",
ORCID-numbers = "Borwein, Jonathan/0000-0002-1263-0646",
}
@Article{Borwein:2015:PPB,
author = "Jonathan M. Borwein and Scott T. Chapman",
title = "{I} Prefer Pi: A Brief History and Anthology of
Articles in the {American Mathematical Monthly}",
journal = j-AMER-MATH-MONTHLY,
volume = "122",
number = "3",
pages = "195--216",
month = mar,
year = "2015",
CODEN = "AMMYAE",
DOI = "https://doi.org/10.4169/amer.math.monthly.122.03.195",
ISSN = "0002-9890 (print), 1930-0972 (electronic)",
ISSN-L = "0002-9890",
bibdate = "Wed Jun 10 09:05:28 MDT 2015",
bibsource = "http://www.jstor.org/stable/10.4169/amermathmont.122.issue-03;
https://www.math.utah.edu/pub/tex/bib/amermathmonthly2010.bib;
https://www.math.utah.edu/pub/tex/bib/pi.bib",
note = "See addenda \cite{Borwein:2015:PPA}.",
URL = "http://www.jstor.org/stable/10.4169/amer.math.monthly.122.03.195",
acknowledgement = ack-nhfb,
author-dates = "Jonathan Michael Borwein (20 May 1951--2 August
2016)",
fjournal = "American Mathematical Monthly",
journal-URL = "http://www.jstor.org/journals/00029890.html",
}
@Book{Cheng:2015:HBP,
author = "Eugenia Cheng",
title = "How to bake $ \pi $: an edible exploration of the
mathematics of mathematics",
publisher = pub-BASIC-BOOKS,
address = pub-BASIC-BOOKS:adr,
pages = "288 (est.)",
year = "2015",
ISBN = "0-465-05171-5 (hardcover), 0-465-05169-3 (e-book)",
ISBN-13 = "978-0-465-05171-7 (hardcover), 978-0-465-05169-4
(e-book)",
LCCN = "QA9 .C4862 2015",
bibdate = "Wed Jun 3 08:27:16 MDT 2015",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib;
z3950.loc.gov:7090/Voyager",
acknowledgement = ack-nhfb,
tableofcontents = "Prolog / 1 \\
Part I: Math / 5\\
1: What is math? / 7 \\
2: Abstraction / 15 \\
3: Principles / 45 \\
4: Process / 57 \\
5: Generalization / 71 \\
6: Internal vs. external / 97 \\
7: Axiomatization / 115 \\
8: What mathematics is / 141 \\
Part II: Category theory / 157 \\
9: What is category theory? / 159 \\
10: Context / 165 \\
11: Relationships / 183 \\
12: Structure / 205 \\
13: Sameness / 221 \\
14: Universal properties / 239 \\
15: What category theory is / 263 \\
Acknowledgments / 281 \\
Index / 283",
}
@Article{Friedmann:2015:QMD,
author = "Tamar Friedmann and C. R. Hagen",
title = "Quantum mechanical derivation of the {Wallis} formula
for $ \pi $",
journal = j-J-MATH-PHYS,
volume = "56",
number = "11",
pages = "112101",
month = nov,
year = "2015",
CODEN = "JMAPAQ",
DOI = "https://doi.org/10.1063/1.4930800",
ISSN = "0022-2488 (print), 1089-7658 (electronic), 1527-2427",
ISSN-L = "0022-2488",
bibdate = "Fri Nov 27 18:09:07 MST 2015",
bibsource = "https://www.math.utah.edu/pub/tex/bib/jmathphys2015.bib;
https://www.math.utah.edu/pub/tex/bib/pi.bib",
note = "See news story \cite{Meyers:2015:NDP}.",
acknowledgement = ack-nhfb,
fjournal = "Journal of Mathematical Physics",
journal-URL = "http://jmp.aip.org/",
}
@Misc{Meyers:2015:NDP,
author = "Catherine Meyers",
title = "New Derivation of Pi Links Quantum Physics and Pure
Math: Researchers stumbled upon a famous
pre-{Newtonian} formula for pi while computing the
energy levels of a hydrogen atom",
day = "10",
month = nov,
year = "2015",
bibdate = "Fri Nov 27 17:45:06 2015",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
note = "See \cite{Friedmann:2015:QMD}",
URL = "https://publishing.aip.org/publishing/journal-highlights/new-derivation-pi-links-quantum-physics-and-pure-math",
acknowledgement = ack-nhfb,
}
@Article{Tracy:2015:OCC,
author = "Suzanne Tracy",
title = "Once-in-a-Century: Celebrating 10 Digits of Pi on
3.14.15 at 9:26:53",
journal = j-SCI-COMPUT,
day = "14",
month = mar,
year = "2015",
CODEN = "SCHRCU",
ISSN = "1930-5753 (print), 1930-6156 (electronic)",
ISSN-L = "1930-5753",
bibdate = "Fri Mar 13 10:17:12 2015",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
URL = "http://www.scientificcomputing.com/blogs/2015/03/once-century-celebrating-10-digits-pi-31415-92653",
acknowledgement = ack-nhfb,
fjournal = "Scientific Computing",
journal-URL = "http://digital.scientificcomputing.com/scientificcomputing/",
}
@Article{Wardhaugh:2015:LCC,
author = "Benjamin Wardhaugh",
title = "A `lost' chapter in the calculation of $ \pi $: {Baron
Zach} and {MS Bodleian 949}",
journal = j-HIST-MATH,
volume = "42",
number = "3",
pages = "343--351",
month = aug,
year = "2015",
CODEN = "HIMADS",
ISSN = "0315-0860 (print), 1090-249X (electronic)",
ISSN-L = "0315-0860",
bibdate = "Tue Aug 4 08:02:12 MDT 2015",
bibsource = "https://www.math.utah.edu/pub/tex/bib/histmath.bib;
https://www.math.utah.edu/pub/tex/bib/pi.bib",
URL = "http://www.sciencedirect.com/science/article/pii/S031508601500018X",
abstract = "The Bodleian library holds a manuscript containing
mathematical tables and a calculation of $ \pi $ to 154
decimal places, last described (in part) in 1802. This
paper provides an outline of the manuscript's contents
and relates it to contemporary developments in the
computation of $ \pi $.",
acknowledgement = ack-nhfb,
fjournal = "Historia Mathematica",
journal-URL = "http://www.sciencedirect.com/science/journal/03150860/",
}
@Article{Bailey:2016:RCS,
author = "David H. Bailey and Jonathan M. Borwein and Richard P.
Brent and Mohsen Reisi",
title = "Reproducibility in Computational Science: A Case
Study: Randomness of the Digits of Pi",
journal = j-EXP-MATH,
volume = "22",
number = "??",
pages = "1--8",
month = "",
year = "2016",
CODEN = "????",
DOI = "https://doi.org/10.1080/10586458.2016.1163755",
ISSN = "1058-6458 (print), 1944-950X (electronic)",
ISSN-L = "1058-6458",
bibdate = "Fri Aug 12 07:22:53 2016",
bibsource = "https://www.math.utah.edu/pub/bibnet/authors/b/borwein-jonathan-m.bib;
https://www.math.utah.edu/pub/tex/bib/pi.bib",
note = "See \cite{Ganz:2014:DES}.",
URL = "http://www.tandfonline.com/doi/full/10.1080/10586458.2016.1163755",
acknowledgement = ack-nhfb,
author-dates = "Jonathan Michael Borwein (20 May 1951--2 August
2016)",
fjournal = "Experimental Mathematics",
journal-URL = "http://www.tandfonline.com/loi/uexm20",
onlinedate = "24 August 2016",
ORCID-numbers = "Bailey, David H./0000-0002-7574-8342; Borwein,
Jonathan/0000-0002-1263-0646",
remark = "This paper reproduces work in \cite{Ganz:2014:DES},
and then shows that the report in that paper of
nonrandomness of digits of $ \pi $ is an artifact of an
unlucky choice of sample block sizes. Statistics from
several different block sizes support the widely-held,
but still unproven, belief that $ \pi $ is a normal
number (where all digits occur with equal
probability).",
xxauthor = "David H. Bailey and Jonathan M. Borwein and Richard P.
Brent and Mohsen Reisi Ardali",
}
@Article{Roberts:2016:HFB,
author = "Gareth Ffowc Roberts",
title = "How a Farm Boy from {Wales} Gave the World Pi",
journal = "Scientific Computing",
volume = "??",
number = "??",
pages = "??",
day = "14",
month = mar,
year = "2016",
bibdate = "Sat Mar 19 09:54:09 2016",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
URL = "http://www.scientificcomputing.com/articles/2016/03/how-farm-boy-wales-gave-world-pi",
acknowledgement = ack-nhfb,
keywords = "William Jones (1674--17??), Leonard Euler
(1707--1783)",
}
@Misc{Bailey:2017:PCP,
author = "David H. Bailey",
title = "Pi and the collapse of peer review",
howpublished = "Web blog.",
day = "20",
month = jul,
year = "2017",
bibdate = "Tue Jul 25 19:00:52 2017",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
URL = "http://mathscholar.org/pi-and-the-collapse-of-peer-review",
acknowledgement = ack-nhfb,
ORCID-numbers = "Bailey, David H./0000-0002-7574-8342",
remark = "Includes remarks on several recently-published papers
with nonsensical claims that $ \pi $ is equal to simple
numerical expressions involving square roots of
integers.",
}
@InCollection{Richeson:2017:CRW,
author = "David Richeson",
title = "Circular reasoning: who first proved that {$C$}
divided by $d$ is a constant?",
crossref = "Pitici:2017:BWM",
pages = "??--??",
year = "2017",
bibdate = "Tue Nov 20 10:49:22 2018",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
acknowledgement = ack-nhfb,
}
@Misc{Yee:2017:PNL,
author = "Alexander Yee",
title = "Pi: Notable Large Computations",
howpublished = "Web blog and tables.",
day = "17",
month = may,
year = "2017",
bibdate = "Tue Jul 25 18:56:30 2017",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
note = "The latest record set on 11 November 2016 by Peter
Trueb is 22,459,157,718,361 decimal digits of $ \pi
$.",
acknowledgement = ack-nhfb,
}
@Misc{Anonymous:2018:BF,
author = "Anonymous",
title = "{Bellard}'s formula",
howpublished = "Wikipedia article",
year = "2018",
bibdate = "Wed Dec 05 14:08:24 2018",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
URL = "https://en.wikipedia.org/wiki/Bellard%27s_formula",
acknowledgement = ack-nhfb,
remark = "From the article: ``It is a faster version (about 43\%
faster) of the Bailey--Borwein--Plouffe formula.''",
}
@Article{Bertot:2018:DDP,
author = "Yves Bertot and Laurence Rideau and Laurent
Th{\'e}ry",
title = "Distant Decimals of $ \pi $: Formal Proofs of Some
Algorithms Computing Them and Guarantees of Exact
Computation",
journal = j-J-AUTOM-REASON,
volume = "61",
number = "1--4",
pages = "33--71",
month = jun,
year = "2018",
CODEN = "JAREEW",
DOI = "https://doi.org/10.1007/s10817-017-9444-2",
ISSN = "0168-7433 (print), 1573-0670 (electronic)",
ISSN-L = "0168-7433",
bibdate = "Sat Aug 4 07:51:41 MDT 2018",
bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib;
https://www.math.utah.edu/pub/tex/bib/jautomreason.bib;
https://www.math.utah.edu/pub/tex/bib/pi.bib",
URL = "http://link.springer.com/article/10.1007/s10817-017-9444-2",
acknowledgement = ack-nhfb,
fjournal = "Journal of Automated Reasoning",
journal-URL = "http://link.springer.com/journal/10817",
keywords = "Arithmetic geometric means; Bailey, Borwein, and
Plouffe formula; BBP; Coq proof assistant; Formal
proofs in real analysis; PI",
}
@Article{Takahashi:2018:CQH,
author = "Daisuke Takahashi",
title = "Computation of the 100 quadrillionth hexadecimal digit
of $ \pi $ on a cluster of {Intel Xeon Phi}
processors",
journal = j-PARALLEL-COMPUTING,
volume = "75",
number = "??",
pages = "1--10",
month = jul,
year = "2018",
CODEN = "PACOEJ",
DOI = "https://doi.org/10.1016/j.parco.2018.02.002",
ISSN = "0167-8191 (print), 1872-7336 (electronic)",
ISSN-L = "0167-8191",
bibdate = "Mon May 14 07:57:43 MDT 2018",
bibsource = "https://www.math.utah.edu/pub/tex/bib/parallelcomputing.bib;
https://www.math.utah.edu/pub/tex/bib/pi.bib",
URL = "http://www.sciencedirect.com/science/article/pii/S0167819118300334",
acknowledgement = ack-nhfb,
fjournal = "Parallel Computing",
journal-URL = "http://www.sciencedirect.com/science/journal/01678191",
}
@Misc{Bailey:2019:SPA,
author = "David H. Bailey",
title = "Simple proofs: {Archimedes}' calculation of pi",
howpublished = "Web site.",
day = "9",
month = feb,
year = "2019",
bibdate = "Tue Apr 21 16:18:17 2020",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
URL = "https://mathscholar.org/2019/02/simple-proofs-archimedes-calculation-of-pi/",
acknowledgement = ack-nhfb,
ORCID-numbers = "Bailey, David H./0000-0002-7574-8342",
}
@Misc{Porter:2019:GEC,
author = "Jon Porter",
title = "{Google} employee calculates pi to record 31 trillion
digits: But remember, only 40 or so of them are
actually useful",
howpublished = "Web site",
day = "14",
month = mar,
year = "2019",
bibdate = "Tue Apr 30 07:55:47 2019",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
URL = "https://www.theverge.com/2019/3/14/18265358/pi-calculation-record-31-trillion-google",
abstract = "A Google employee from Japan has set a new world
record for the number of digits of pi calculated. Emma
Haruka Iwao, who works as a cloud developer advocate at
Google, calculated pi to 31,415,926,535,897 digits,
smashing the previous record of 22,459,157,718,361
digits set back in 2016. Although Iwao was using the
same y-cruncher program to calculate pi as the previous
record holder, her advantage lay in the use of Google's
cloud-based compute engine. The 31 trillion digits of
pi took 25 virtual machines 121 days to calculate. In
contrast, the previous record holder, Peter Trueb, used
just a single fast computer, albeit one equipped with
two dozen 6TB hard drives to handle the huge dataset
that was produced. His calculation only took 105 days
to complete. Outside of bragging rights, the 9 trillion
extra digits are unlikely to have too many real-world
uses. NASA only uses around 15 digits of pi to send
rockets into space, and measuring the visible
Universe's circumference to the precision of a single
atom would take just 40 digits.",
acknowledgement = ack-nhfb,
}
@TechReport{Bailey:2020:CMF,
author = "David H. Bailey",
title = "A catalogue of mathematical formulas involving $ \pi
$, with analysis",
type = "Report",
institution = "Lawrence Berkeley National Laboratory, and Department
of Computer Science, University of California, Davis",
address = "Berkeley, CA 94720 and Davis, CA",
pages = "14",
day = "27",
month = mar,
year = "2020",
bibdate = "Tue Apr 21 13:02:03 2020",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
URL = "https://www.davidhbailey.com/dhbpapers/pi-formulas.pdf",
abstract = "This paper presents a catalogue of mathematical
formulas and iterative algorithms for evaluating the
mathematical constant $ \pi $, ranging from Archimedes'
2200-year-old iteration to some formulas that were
discovered only in the past few decades. Computer
implementations and timing results for these formulas
and algorithms are also included. In particular,
timings are presented for evaluations of various
infinite series formulas to approximately 10,000-digit
precision, for evaluations of various integral formulas
to approximately 4,000-digit precision, and for
evaluations of several iterative algorithms to
approximately 100,000-digit precision, all based on
carefully designed comparative computer runs.",
acknowledgement = ack-nhfb,
ORCID-numbers = "Bailey, David H./0000-0002-7574-8342",
remark = "Three minor typos fixed on 22 April 2020, and document
redated.",
}
@InProceedings{Brent:2020:BBP,
author = "Richard P. Brent",
title = "The {Borwein} Brothers, Pi and the {AGM}",
crossref = "Bailey:2020:AVC",
pages = "323--347",
year = "2020",
DOI = "https://doi.org/10.1007/978-3-030-36568-4_21",
bibdate = "Tue Apr 21 10:54:18 MDT 2020",
bibsource = "https://www.math.utah.edu/pub/bibnet/authors/b/borwein-jonathan-m.bib;
https://www.math.utah.edu/pub/tex/bib/agm.bib;
https://www.math.utah.edu/pub/tex/bib/pi.bib",
acknowledgement = ack-nhfb,
subject-dates = "Jonathan Michael Borwein (20 May 1951--2 August
2016)",
}
@Article{Monroe:2020:NBB,
author = "Don Monroe",
title = "News: Bouncing balls and quantum computing",
journal = j-CACM,
volume = "63",
number = "10",
pages = "10--12",
month = sep,
year = "2020",
CODEN = "CACMA2",
DOI = "https://doi.org/10.1145/3416076",
ISSN = "0001-0782 (print), 1557-7317 (electronic)",
ISSN-L = "0001-0782",
bibdate = "Thu Sep 24 07:02:28 MDT 2020",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib;
https://www.math.utah.edu/pub/tex/bib/cacm2020.bib",
URL = "https://dl.acm.org/doi/10.1145/3416076",
abstract = "A lighthearted method for calculating
$ \pi $ is analogous to a fundamental
algorithm for quantum computing.",
acknowledgement = ack-nhfb,
fjournal = "Communications of the ACM",
journal-URL = "https://dl.acm.org/loi/cacm",
}
@Misc{Yee:2020:CMT,
author = "Alexander J. Yee",
title = "{{\tt y-cruncher}}: a multi-threaded pi-program",
howpublished = "Web site",
day = "30",
month = mar,
year = "2020",
bibdate = "Tue Apr 21 16:09:31 2020",
bibsource = "https://www.math.utah.edu/pub/tex/bib/multithreading.bib;
https://www.math.utah.edu/pub/tex/bib/pi.bib",
URL = "http://www.numberworld.org/y-cruncher/;
https://www.fhgr.ch/en/specialist-areas/applied-future-technologies/davis-centre/pi-challenge/;
https://www.hpcwire.com/2021/08/19/after-108-days-swiss-hpc-system-calculates-pi-to-62-8-trillion-digits/",
abstract = "How fast can your computer compute Pi?\par
y-cruncher is a program that can compute Pi and other
constants to trillions of digits.\par
It is the first of its kind that is multi-threaded and
scalable to multi-core systems. Ever since its launch
in 2009, it has become a common benchmarking and
stress-testing application for overclockers and
hardware enthusiasts.\par
y-cruncher has been used to set several world records
for the most digits of Pi ever computed:\par
62.8 trillion digits - August 2021 (Thomas Keller and
Heiko R{\"o}lke, Graub{\"u}nden University of Applied
Sciences, Switzerland)
50 trillion digits - January 2020 (Timothy
Mullican)\par
31.4 trillion digits - January 2019 (Emma Haruka
Iwao)\par
22.4 trillion digits - November 2016 (Peter
Trueb)\par
13.3 trillion digits - October 2014 (Sandon Van Ness
``houkouonchi'')\par
12.1 trillion digits - December 2013 (Shigeru
Kondo)\par
10 trillion digits - October 2011 (Shigeru Kondo)\par
5 trillion digits - August 2010 (Shigeru Kondo)",
acknowledgement = ack-nhfb,
}
@Article{Staudte:2020:EGF,
author = "R. G. Staudte",
title = "Evidence for goodness of fit in {Karl Pearson}
chi-squared statistics",
journal = j-STATISTICS,
volume = "54",
number = "6",
pages = "1287--1310",
year = "2020",
CODEN = "MOSSD5",
DOI = "https://doi.org/10.1080/02331888.2020.1862115",
ISSN = "0233-1888 (print), 1029-4910 (electronic)",
ISSN-L = "0233-1888",
bibdate = "Tue May 18 10:50:09 MDT 2021",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib;
https://www.math.utah.edu/pub/tex/bib/prng.bib;
https://www.math.utah.edu/pub/tex/bib/statistics.bib",
abstract = "Chi-squared tests for lack of fit are traditionally
employed to find evidence against a hypothesized model,
with the model accepted if the Karl Pearson statistic
comparing observed and expected numbers of observations
falling within cells is not significantly large.
However, if one really wants evidence for goodness of
fit, it is better to adopt an equivalence testing
approach in which small values of the chi-squared
statistic indicate evidence for the desired model. This
method requires one to define what is meant by
equivalence to the desired model, and guidelines are
proposed. It is shown that the evidence for equivalence
can distinguish between normal and nearby models, as
well between the Poisson and over-dispersed models.
Applications to the evaluation of random number
generators and to uniformity of the digits of pi are
included. Sample sizes required to obtain a desired
expected evidence for goodness of fit are also
provided.",
acknowledgement = ack-nhfb,
fjournal = "Statistics: a Journal of Theoretical and Applied
Statistics",
journal-URL = "http://www.tandfonline.com/loi/gsta20",
onlinedate = "23 Dec 2020",
}
@Article{Guillera:2021:PRR,
author = "Jes{\'u}s Guillera",
title = "Proof of a rational {Ramanujan}-type series for $ 1 /
\pi $. {The} fastest one in level 3",
journal = j-INT-J-NUMBER-THEORY,
volume = "17",
number = "02",
pages = "473--477",
month = mar,
year = "2021",
DOI = "https://doi.org/10.1142/S1793042120400242",
ISSN = "1793-0421 (print), 1793-7310 (electronic)",
ISSN-L = "1793-0421",
bibdate = "Tue May 18 16:16:01 MDT 2021",
bibsource = "https://www.math.utah.edu/pub/tex/bib/ijnt.bib;
https://www.math.utah.edu/pub/tex/bib/pi.bib",
URL = "https://www.worldscientific.com/doi/10.1142/S1793042120400242",
abstract = "Using a modular equation of level 3 and degree 23 due
to Chan and Liaw, we prove the fastest known
(conjectured to be the fastest one) convergent rational
Ramanujan-type series for 1 \/ \pi of level 3 .",
acknowledgement = ack-nhfb,
fjournal = "International Journal of Number Theory (IJNT)",
journal-URL = "https://www.worldscientific.com/worldscinet/ijnt",
remark = "Special Issue I: In Honor of Bruce Berndt's 80th
Birthday",
}
@Article{Ernstsson:2022:DPP,
author = "August Ernstsson and Nicolas Vandenbergen and
Christoph Kessler",
title = "A Deterministic Portable Parallel Pseudo-Random Number
Generator for Pattern-Based Programming of
Heterogeneous Parallel Systems",
journal = j-INT-J-PARALLEL-PROG,
volume = "50",
number = "3-4",
pages = "319--340",
month = aug,
year = "2022",
CODEN = "IJPPE5",
DOI = "https://doi.org/10.1007/s10766-022-00726-5",
ISSN = "0885-7458 (print), 1573-7640 (electronic)",
ISSN-L = "0885-7458",
bibdate = "Fri Jul 15 17:25:07 MDT 2022",
bibsource = "https://www.math.utah.edu/pub/tex/bib/intjparallelprogram.bib;
https://www.math.utah.edu/pub/tex/bib/pi.bib;
https://www.math.utah.edu/pub/tex/bib/prng.bib",
URL = "https://link.springer.com/article/10.1007/s10766-022-00726-5",
acknowledgement = ack-nhfb,
ajournal = "Int. J. Parallel Prog.",
fjournal = "International Journal of Parallel Programming",
journal-URL = "http://link.springer.com/journal/10766",
remark = "This article uses the Buffon needle-casting algorithm
to estimate the numerical value of $ \pi $ in
sequential and parallel versions, but with the
guarantee that both produce identical results.",
}
@Article{Lucas:2022:MSF,
author = "Stephen K. Lucas and Amrik Singh Nimbran",
title = "Monotonic series for fractions near $ \pi $ and their
convergents",
journal = j-MATH-GAZ,
volume = "106",
number = "566",
pages = "300--309",
month = jul,
year = "2022",
CODEN = "MAGAAS",
DOI = "https://doi.org/10.1017/mag.2022.70",
ISSN = "0025-5572 (print), 2056-6328 (electronic)",
ISSN-L = "0025-5572",
bibdate = "Mon Jul 18 07:47:31 MDT 2022",
bibsource = "https://www.math.utah.edu/pub/tex/bib/mathgaz2020.bib;
https://www.math.utah.edu/pub/tex/bib/pi.bib",
URL = "https://www.cambridge.org/core/journals/mathematical-gazette/article/monotonic-series-for-fractions-near-and-their-convergents/73CAD6F4E882A4643E7C626066EA5891",
acknowledgement = ack-nhfb,
ajournal = "Math. Gaz.",
fjournal = "The Mathematical Gazette",
journal-URL = "http://journals.cambridge.org/action/displayIssue?jid=MAG;
http://www.m-a.org.uk/jsp/index.jsp?lnk=620",
onlinedate = "22 June 2022",
}
@Misc{Yee:2022:CMT,
author = "Alexander J. Yee",
title = "{{\tt y-cruncher}} --- a multi-threaded pi-program",
howpublished = "Web site",
day = "13",
month = oct,
year = "2022",
bibdate = "Mon Dec 05 08:24:08 2022",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
URL = "http://www.numberworld.org/y-cruncher/",
abstract = "y-cruncher is a program that can compute Pi and other
constants to trillions of digits. It is the first of
its kind that is multi-threaded and scalable to
multi-core systems. Ever since its launch in 2009, it
has become a common benchmarking and stress-testing
application for overclockers and hardware
enthusiasts.",
acknowledgement = ack-nhfb,
remark = "From the Web site:\\
100 trillion digits - June 2022 (Emma Haruka Iwao) \\
62.8 trillion digits - August 2021 (UAS Grisons) \\
50 trillion digits - January 2020 (Timothy Mullican)
\\
31.4 trillion digits - January 2019 (Emma Haruka Iwao)
\\
22.4 trillion digits - November 2016 (Peter Trueb) \\
13.3 trillion digits - October 2014 (Sandon Van Ness
`houkouonchi'') \\
12.1 trillion digits - December 2013 (Shigeru Kondo)
\\
10 trillion digits - October 2011 (Shigeru Kondo) \\
5 trillion digits - August 2010 (Shigeru Kondo)",
}
@Misc{Bailey:2023:CBT,
author = "David H. Bailey",
title = "A Compendium of {BBP}-Type Formulas for Mathematical
Constants",
day = "8",
month = apr,
year = "2023",
bibdate = "Mon Apr 17 17:55:20 2023",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
note = "Web report.",
URL = "https://www.davidhbailey.com/dhbpapers/bbp-formulas.pdf",
abstract = "A 1996 paper by the author, Peter Borwein and Simon
Plouffe showed that any mathematical constant given by
an infinite series of a certain type has the property
that its n-th digit in a particular number base could
be calculated directly, without needing to compute any
of the first n 1 digits, by means of a simple algorithm
that does not require multiple-precision arithmetic.
Several such formulas were presented in that paper,
including formulas for the constants and log 2. Since
then, numerous other formulas of this type have been
found. This paper presents a compendium of currently
known results of this sort, together with citations and
references.",
acknowledgement = ack-nhfb,
remark = "This paper reports a new finding that Leohard Euler
found two such formulas in 1779, but their significance
was not recognized for more than two centuries; see
\cite{Craig-Wood:2023:EFF}.",
}
@Article{Craig-Wood:2023:EFF,
author = "Nick Craig-Wood",
title = "{Euler} Found the First Binary Digit Extraction
Formula for $ \pi $ in 1779",
journal = j-EULERIANA,
volume = "3",
number = "1",
pages = "23--30",
month = mar,
year = "2023",
CODEN = "????",
DOI = "https://doi.org/10.56031/2693-9908.1049",
ISSN = "2693-9908",
bibdate = "Tue Apr 18 09:56:27 MDT 2023",
bibsource = "https://www.math.utah.edu/pub/tex/bib/euleriana.bib;
https://www.math.utah.edu/pub/tex/bib/pi.bib",
URL = "https://scholarlycommons.pacific.edu/euleriana/vol3/iss1/3",
abstract = "In 1779 Euler discovered two formulas for $ \pi $
which can be used to calculate any binary digit of $
\pi $ without calculating the previous digits. Up until
now it was believed that the first formula with the
correct properties (known as a BBP-type formula) for
this calculation was published by Bailey, Borwein and
Plouffe in 1997.",
acknowledgement = ack-nhfb,
articleno = "3",
fjournal = "Euleriana",
journal-URL = "https://scholarlycommons.pacific.edu/euleriana/",
}
@Article{Strickland:2023:HLT,
author = "Lloyd Strickland",
title = "How {Leibniz} tried to tell the world he had squared
the circle",
journal = j-HIST-MATH,
volume = "62",
number = "??",
pages = "19--39",
month = feb,
year = "2023",
CODEN = "HIMADS",
DOI = "https://doi.org/10.1016/j.hm.2022.08.004",
ISSN = "0315-0860 (print), 1090-249X (electronic)",
ISSN-L = "0315-0860",
bibdate = "Wed Mar 15 09:40:56 MDT 2023",
bibsource = "https://www.math.utah.edu/pub/tex/bib/histmath.bib;
https://www.math.utah.edu/pub/tex/bib/pi.bib",
URL = "http://www.sciencedirect.com/science/article/pii/S0315086022000465",
acknowledgement = ack-nhfb,
fjournal = "Historia Mathematica",
journal-URL = "http://www.sciencedirect.com/science/journal/03150860",
remark = "A Maple calculation of Liebniz's sum is
evalf(add((-1)**k * (1 / (2*k + 1)), k = 0..1e6), 30) /
(Pi * (1/2)**2) returns 1.000000318, which is on the
right track, but certainly not a fast computational
route to a numerical approximation to $\pi$!",
}
%%% ====================================================================
%%% Cross-referenced entries must come last:
@Proceedings{Traub:1976:ACC,
editor = "J. F. (Joseph Frederick) Traub",
booktitle = "{Analytic computational complexity: Proceedings of the
Symposium on Analytic Computational Complexity, held by
the Computer Science Department, Carnegie-Mellon
University, Pittsburgh, Pennsylvania, on April 7--8,
1975}",
title = "{Analytic computational complexity: Proceedings of the
Symposium on Analytic Computational Complexity, held by
the Computer Science Department, Carnegie-Mellon
University, Pittsburgh, Pennsylvania, on April 7--8,
1975}",
publisher = pub-ACADEMIC,
address = pub-ACADEMIC:adr,
pages = "ix + 239",
year = "1976",
ISBN = "0-12-697560-4",
ISBN-13 = "978-0-12-697560-4",
LCCN = "QA297 .S915 1975",
bibdate = "Sun Dec 30 18:48:22 MST 2007",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib;
z3950.loc.gov:7090/Voyager",
acknowledgement = ack-nhfb,
meetingname = "Symposium on Analytic Computational Complexity,
Carnegie-Mellon University, 1975.",
remark = "",
subject = "Numerical analysis; Data processing; Congresses;
Computational complexity",
}
@Proceedings{Singh:1984:ATS,
editor = "S. P. Singh and J. W. H. Burry and B. Watson",
booktitle = "{Approximation Theory and Spline Functions. NATO
Advanced Study Institute held at Memorial University of
Newfoundland during August 22--September 2, 1983}",
title = "{Approximation Theory and Spline Functions. NATO
Advanced Study Institute held at Memorial University of
Newfoundland during August 22--September 2, 1983}",
volume = "136",
publisher = pub-SV,
address = pub-SV:adr,
pages = "ix + 485",
year = "1984",
DOI = "https://doi.org/10.1007/978-94-009-6466-2",
ISBN = "94-009-6466-8, 94-009-6468-4",
ISBN-13 = "978-94-009-6466-2, 978-94-009-6468-6",
ISSN = "1389-2185",
LCCN = "????",
bibdate = "Tue Aug 16 11:51:58 MDT 2016",
bibsource = "fsz3950.oclc.org:210/WorldCat;
https://www.math.utah.edu/pub/bibnet/authors/b/borwein-jonathan-m.bib;
https://www.math.utah.edu/pub/tex/bib/pi.bib",
series = "NATO ASI Series, Series C: Mathematical and Physical
Sciences",
acknowledgement = ack-nhfb,
remark = "A NATO Advanced Study Institute on Approximation
Theory and Spline Functions was held at Memorial
University of Newfoundland during August 22--September
2, 1983. This volume consists of the Proceedings of
that Institute. These Proceedings include the main
invited talks and contributed papers given during the
Institute. The aim of these lectures was to bring
together Mathematicians, Physicists and Engineers
working in the field. The lectures covered a wide range
including Multivariate Approximation, Spline Functions,
Rational Approximation, Applications of Elliptic
Integrals and Functions in the Theory of Approximation,
and Pad{\'e} Approximation. We express our sincere
thanks to Professors E. W. Cheney, J. Meinguet, J. M.
Phillips and H. Werner, members of the International
Advisory Committee. We also extend our thanks to the
main speakers and the invi ted speakers, whose
contributions made these Proceedings complete. The
Advanced Study Institute was financed by the NATO
Scientific Affairs Division. We express our thanks for
the generous support. We wish to thank members of the
Department of Mathematics and Statistics at Memorial
University who willingly helped with the planning and
organizing of the Institute. Special thanks go to Mrs.
Mary Pike who helped immensely in the planning and
organizing of the Institute, and to Miss Rosalind Genge
for her careful and excellent typing of the manuscript
of these Proceedings.",
subject = "Analysis; Approximations and Expansions; Global
analysis (Mathematics); Mathematics",
tableofcontents = "Front Matter / i--ix \\
Products of Polynomials / Bernard Beauzamy / 1--22 \\
Exchange Algorithms, Error Estimations and Strong
Unicity in Convex Programming and Chebyshev
Approximation / Hans-Peter Blatt / 23--63 \\
Four Lectures on Multivariate Approximation / E. W.
Cheney / 65--87 \\
The Approximation of Certain Functions by Compound
Means / D. M. E. Foster, G. M. Phillips / 89--95 \\
A Practical Method for Obtaining a Priori Error Bounds
in Pointwise and Mean-Square Approximation Problems /
Jean Meinguet / 97--125 \\
Surface Spline Interpolation: Basic Theory and
Computational Aspects / Jean Meinguet / 127--142 \\
Interpolation of Scattered Data: Distance Matrices and
Conditionally Positive Definite Functions / Charles A.
Micchelli / 143--145 \\
Semi-Norms in Polynomial Approximation / G. M.
Phillips, P. J. Taylor / 147--150 \\
On Spaces of Piecewise Polynomials in Two Variables /
Larry L. Schumaker / 151--197 \\
Birkhoff Interpolation on the Roots of Unity / A.
Sharma / 199--205 \\
Applications of Transformation Theory: A Legacy from
Zolotarev (1847--1878) / John Todd / 207--245 \\
Explicit Algebraic Nth Order Approximations to PI / J.
M. Borwein, P. B. Borwein / 247--256 \\
Solving Integral Equations of Nuclear Scattering by
Splines / M. Brannigan / 257--264 \\
$H$-Sets for Non-Linear Constrained Approximation / M.
Brannigan / 265--270 \\
Operator Pad{\'e} Approximants: Some ideas behind the
theory and a numerical illustration / Annie A. H. Cuyt
/ 271--288 \\
Harmonic Approximation / Myron Goldstein / 289--292 \\
Best Harmonic L1 Approximation to Subharmonic Functions
/ M. Goldstein, W. Haussman, K. Jetter / 293--295 \\
B-Splines on the Circle and Trigonometric B-Splines /
T. N. T. Goodman, S. L. Lee / 297--325 \\
On Reducing the Computational Error in the Successive
Approximations Method / Fran{\c{c}}ois B. Gu{\'e}nard /
327--338 \\
Lebesgue Constants Determined by Extremal Sets / Myron
S. Henry / 339--348 \\
Error Bounds for Interpolation by Fourth Order
Trigonometric Splines / P. E. Koch / 349--360 \\
Approximation of Derivatives in $\mathbb{R}^n$
Application: Construction of Surfaces in $\mathbb{R}^2$
/ Alain Le Mehaute / 361--378 \\
Meromorphic Functions, Maps and Their Rational
Approximants in $\mathbb{C}^n$ / C. H. Lutterodt /
379--396 \\
Splines and Collocation for Ordinary Initial Value
Problems / Syvert P. Norsett / 397--417 \\
Degree of Approximation of Quasi-Hermite--Fej{\'e}r
Interpolation Based on Jacobi Abscissas $P_n(\alpha,
\alpha) (x)$ / J. Prasad, A. K. Varma / 419--440 \\
Using Inclusion Theorems to Establish the Summability
of Orthogonal Series / B. E. Rhoades / 441--453 \\
On Projections in Approximation Theory / Boris
Shekhtman / 455--466 \\
A Survey of Exterior Asymptotics for Orthogonal
Polynomials Associated with a Finite Interval and a
Study of the Case of the General Weight Measures /
Joseph L. Ullman / 467--478 \\ / Back Matter /
479--485",
}
@Proceedings{Monien:1986:SAS,
editor = "B. Monien and G. Vidal-Naquet",
booktitle = "{STACS} 86: 3rd Annual Symposium on Theoretical
Aspects of Computer Science, Orsay, France, January
16--18, 1986",
title = "{STACS} 86: 3rd Annual Symposium on Theoretical
Aspects of Computer Science, Orsay, France, January
16--18, 1986",
volume = "210",
publisher = pub-SV,
address = pub-SV:adr,
pages = "ix + 368",
year = "1986",
CODEN = "LNCSD9",
DOI = "https://doi.org/10.1007/3-540-16078-7",
ISBN = "0-387-16078-7 (paperback)",
ISBN-13 = "978-0-387-16078-8 (paperback)",
ISSN = "0302-9743 (print), 1611-3349 (electronic)",
LCCN = "QA267.A1 L43 no.210",
bibdate = "Fri Apr 12 07:14:49 1996",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib",
note = "Organized jointly by the special interest group for
theoretical computer science of the Gesellschaft
f{\"u}r Informatik (G.I.) and the special interest
group for applied mathematic[s] of the Association
fran{\c{c}}aise des sciences et techniques de
l'information, de l'organisation et des syst{\`e}mes
(AFCET)''",
series = ser-LNCS,
URL = "http://link.springer-ny.com/link/service/series/0558/tocs/t0210.htm;
http://www.springer.com/computer/theoretical+computer+science/book/978-3-540-16078-6;
http://www.springerlink.com/openurl.asp?genre=issue&issn=0302-9743&volume=210",
acknowledgement = ack-nhfb,
keywords = "computers --- congresses; electronic data processing
--- congresses",
}
@Proceedings{Martin:1988:SPN,
editor = "Joanne L. Martin and Stephen F. Lundstrom",
booktitle = "Supercomputing '88: proceedings, November 14--18,
1988, Orlando, Florida",
title = "Supercomputing '88: proceedings, November 14--18,
1988, Orlando, Florida",
volume = "2",
publisher = pub-IEEE,
address = pub-IEEE:adr,
pages = "viii + 263",
year = "1988",
ISBN = "0-8186-0882-X (v. 1; paper), 0-8186-8882-3 (v. 1;
case), 0-8186-4882-1 (v. 1: microfiche) 0-8186-8923-4
(v. 2), 0-8186-5923-X (v. 2: microfiche), 0-8186-8923-4
(v. 2: case)",
ISBN-13 = "978-0-8186-0882-7 (v. 1; paper), 978-0-8186-8882-9 (v.
1; case), 978-0-8186-4882-3 (v. 1: microfiche)
978-0-8186-8923-9 (v. 2), 978-0-8186-5923-2 (v. 2:
microfiche), 978-0-8186-8923-9 (v. 2: case)",
LCCN = "QA76.5 .S894 1988",
bibdate = "Fri Aug 30 08:01:51 MDT 1996",
bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib;
https://www.math.utah.edu/pub/tex/bib/pi.bib",
note = "Two volumes. IEEE catalog number 88CH2617-9. IEEE
Computer Society Order Number 882.",
acknowledgement = ack-nhfb,
classification = "C5440 (Multiprocessor systems and techniques); C7300
(Natural sciences)",
keywords = "biology computing; chemistry; computational biology;
computational fluid dynamics; computational
mathematics; computational physics; flow simulation;
global change; mathematics computing; parallel
processing; physics computing; structural analysis;
structural engineering computing; supercomputers ---
congresses",
}
@Book{Berggren:1997:PSB,
editor = "Lennart Berggren and Jonathan M. Borwein and Peter B.
Borwein",
booktitle = "Pi, a source book",
title = "Pi, a source book",
publisher = pub-SV,
address = pub-SV:adr,
pages = "xix + 716",
year = "1997",
DOI = "https://doi.org/10.1007/978-1-4757-2736-4",
ISBN = "0-387-94924-0, 1-4757-2736-4 (e-book), 1-4757-2738-0
(print), 3-540-94924-0",
ISBN-13 = "978-0-387-94924-6, 978-1-4757-2736-4 (e-book),
978-1-4757-2738-8 (print), 978-3-540-94924-4",
LCCN = "QA484 .P5 1997",
bibdate = "Fri Sep 2 17:41:50 MDT 2022",
bibsource = "fsz3950.oclc.org:210/WorldCat;
https://www.math.utah.edu/pub/bibnet/authors/b/borwein-jonathan-m.bib;
https://www.math.utah.edu/pub/tex/bib/agm.bib;
https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
https://www.math.utah.edu/pub/tex/bib/pi.bib;
z3950.loc.gov:7090/Voyager",
abstract = "The aim of this book is to provide a complete history
of pi from the dawn of mathematical time to the
present. The story of pi reflects the most seminal, the
most serious and sometimes the silliest aspects of
mathematics, and a surprising amount of the most
important mathematics and mathematicians have
contributed to its unfolding. Pi is one of the few
concepts in mathematics whose mention evokes a response
of recognition and interest in those not concerned
professionally with the subject. Yet, despite this, no
source book on pi has been published. One of the
beauties of the literature on pi is that it allows for
the inclusion of very modern, yet still accessible,
mathematics. Mathematicians and historians of
mathematics will find this book indispensable. Teachers
at every level from the seventh grade onward will find
here ample resources for anything from special topic
courses to individual talks and special student
projects. The literature on pi included in this source
book falls into three classes: first a selection of the
mathematical literature of four millennia, second a
variety of historical studies or writings on the
cultural meaning and significance of the number, and
third, a number of treatments on pi that are fanciful,
satirical and/or whimsical.",
acknowledgement = ack-nhfb,
ORCID-numbers = "Borwein, Jonathan/0000-0002-1263-0646",
subject = "Pi; Pi (Le nombre); Pi.; Pi (le nombre)",
tableofcontents = "Preface / v \\
\\
Acknowledgments / ix \\
\\
Introduction / xvii \\
\\
1. The Rhind Mathematical Papyrus-Problem 50 ($\approx$
1650 B.C.) / A problem dealing with the area of a round
field of given diameter / 1 \\
\\
2. Engels. Quadrature of the Circle in Ancient Egypt
(1977) / A conjectural explanation of how the
mathematicians of ancient Egypt approximated the area
of a circle / 3 \\
\\
3. Archimedes. Measurement of a Circle ($\approx$ 250
BC) / The seminal work in which Archimedes presents the
first true algorithm for $\pi$ / 7 \\
\\
4. Phillips. Archimedes the Numerical Analyst (1981) /
A summary of Archimedes' work on the computation of
$\pi$ using modern notation / 15 \\
\\
5. Lam and Ang. Circle Measurements in Ancient China
(1986) / This paper discusses and contains a
translation of Liu Hui's (3rd century) method for
evaluating $\pi$ and also examines values for $\pi$
given by Zu Chongzhi (429--500) / 20 \\
\\
6. The Ban{\=u} M{\=u}s{\=a}: The Measurement of Plane
and Solid Figures ($\approx$ 850) / This extract gives
an explicit statement and proof that the ratio of the
circumference to the diameter is constant / 36 \\
\\
7. M{\=a}dhava. The Power Series for Arctan and Pi
($\approx$ 1400) / These theorems by a fifteenth
century Indian mathematician give Gregory's series for
arctan with remainder terms and Leibniz's series for
$\pi$ / 45 \\
\\
8. Hope-Jones. Ludolph (or Ludolff or Lucius) van
Ceulen (1938) / Correspondence about van Ceulen's
tombstone in reference to it containing some digits of
$\pi$ / 51 \\
\\
9. Vi{\'e}te. Variorum de Rebus Mathematicis Reponsorum
Liber VII (1593) / Two excerpts. One containing the
first infinite expression of $\pi$, obtained by
relating the area of a regular $2n$-gon to that of a
regular $n$-gon / 53 \\
\\
10. Wallis. Computation of $\pi$ by Successive
Interpolations (1655) / How Wallis derived the infinite
product for $\pi$ that bears his name / 68 \\
\\
11. Wallis. Arithmetica Infinitorum (1655) / An excerpt
including Prop. 189, 191 and an alternate form of the
result that gives Wm. Brounker's continued fraction
expression for $4/\pi$ / 78 \\
\\
12. Huygens. De Circuli Magnitudine Inventa (1724) /
Huygens's proof of W. Snell's discovery of improvements
in Archimedes' method of estimating the lengths of
circular arcs / 81 \\
\\
13. Gregory. Correspondence with John Collins (1671) /
A letter to Collins in which he gives his series for
arctangent, carried to the ninth power. / 87 \\
\\
14. Roy. The Discovery of the Series Formula for $\pi$
by Leibniz, Gregory, and Nilakantha (1990) / A
discussion of the discovery of the series $\pi/4 = 1 -
1/3 + 1/5, \cdots{}$ / 92 \\
\\
15. Jones. The First Use of $\pi$ for the Circle Ratio
(1706) / An excerpt from Jones' book, the Synopsis
Palmariorum Matheseos: or, a New Introduction to the
Mathematics, London, 1706 / 108 \\
\\
16. Newton. Of the Method of Fluxions and Infinite
Series (1737) / An excerpt giving Newton's calculation
of $\pi$ to 16 decimal places / 110 \\
\\
17. Euler. Chapter 10 of Introduction to Analysis of
the Infinite (On the Use of the Discovered Fractions to
Sum Infinite Series) (1748) / This includes many of
Euler's infinite series for $\pi$ and powers of $\pi$ /
112 \\
\\
18. Lambert. M{\'e}moire Sur Quelques
Propri{\'e}t{\'e}s Remarquables Des Quantit{\'e}s
Transcendentes Circulaires et Logarithmiques (1761) /
An excerpt from Lambert's original proof of the
irrationality of $\pi$ / 129 \\
\\
19. Lambert. Irrationality of $\pi$ (1969) / A
translation and Struik's discussion of Lambert's proof
of the irrationality of $\pi$ / 141 \\
\\
20. Shanks. Contributions to Mathematics Comprising
Chiefly of the Rectification of the Circle to 607
Places of Decimals (1853) / Pages from Shank's report
of his monumental hand calculation of $\pi$ / 147 \\
\\
21. Hermite. Sur La Fonction Exponentielle (1873) / The
first proof of the transcendence of $e$ / 162 \\
\\
22. Lindemann. Ueber die Zahl $\pi$ (1882) / The first
proof of the transcendence of $\pi$ / 194 \\
\\
23. Weierstrass. Zu Lindemann's Abhandlung ``Uber die
Ludolphsche Zahl'' (1885) / Weierstrass' proof of the
transcendence of $\pi$ / 207 \\
\\
24. Hilbert. Ueber die Trancendenz der Zahlen $e$ und
$\pi$ (1893) / Hilbert's short and elegant
simplification of the transcendence proofs for $e$ and
$\pi$ / 226 \\
\\
25. Goodwin. Quadrature of the Circle (1894) / The
dubious origin of the attempted legislation of the
value of $\pi$ in Indiana / 230 \\
\\
26. Edington. House Bill No. 246, Indiana State
Legislature, 1897 (1935) / A summary of the action
taken by the Indiana State Legislature to fix the value
of $\pi$ (including a copy of the actual bill that was
proposed) / 231 \\
\\
27. Singmaster. The Legal Values of Pi (1985) / A
history of the attempt by Indiana to legislate the
value of $\pi$ / 236 \\
\\
28. Ramanujan. Squaring the Circle (1913) / A geometric
approximation to $\pi$ / 240 \\
\\
29. Ramanujan. Modular Equations and Approximations to
$\pi$ (1914) / Ramanujan's seminal paper on $\pi$ that
includes a number of striking series and algebraic
approximations / 241 \\
\\
30. Watson. The Marquis and the Land Agent: A Tale of
the Eighteenth Century (1933) / A Presidential address
to the Mathematical Association in which the author
gives an account of ``some of the elementary work on
arcs and ellipses and other curves which led up to the
idea of inverting an elliptic integral, and so laying
the foundations of elliptic functions and doubly
periodic functions generally.'' / 258 \\
\\
31. Ballantine. The Best (?) Formula for Computing
$\pi$ to a Thousand Places (1939) / An early attempt to
orchestrate the calculation of $\pi$ more cleverly /
271 \\
\\
32. Birch. An Algorithm for Construction of Arctangent
Relations (1946) / The object of this note is to
express $\pi / 4 $ as a sum of arctan relations in
powers of 10 / 274 \\
\\
33. Niven. A Simple Proof that $\pi$ Is Irrational
(1947) / A very concise proof of the irrationality of
$\pi$ / 276 \\
\\
34. Reitwiesner. An ENIAC Determination of $\pi$ and
$e$ to 2000 Decimal Places (1950) / One of the first
computer-based computations / 277 \\
\\
35. Schepler. The Chronology of Pi (1950) / A fairly
reliable outline of the history of $\pi$ from 3000 BC
to 1949 / 282 \\
\\
36. Mahler. On the Approximation of $\pi$ (1953) /
``The aim of this paper is to determine an explicit
lower bound free of unknown constants for the distance
of $\pi$ from a given rational or algebraic number'' /
306 \\
\\
37. Wrench, Jr. The Evolution of Extended Decimal
Approximations to $\pi$ (1960) / A history of the
calculation of the digits of $\pi$ to 1960 \\
\\
38. Shanks and Wrench, Jr. Calculation of $\pi$ to
100,000 Decimals (1962) / A landmark computation of
$\pi$ to more than 100,000 places / 326 \\
\\
39. Sweeny. On the Computation of Euler's Constant
(1963) / The computation of Euler's constant to 3566
decimal places / 350 \\
\\
40. Baker. Approximations to the Logarithms of Certain
Rational Numbers (1964) / The main purpose of this deep
and fundamental paper is to ``deduce results concerning
the accuracy with which the natural logarithms of
certain rational numbers may be approximated by
rational numbers, or, more generally, by algebraic
numbers of bounded degree.'' / 359 \\
\\
41. Adams. Asymptotic Diophantine Approximations to $E$
(1966) / An asymptotic estimate for the rational
approximation to $e$ which disproves the conjecture
that $e$ behaves like almost all numbers in this
respect / 368 \\
\\
42. Mahler. Applications of Some Formulae by Hermite to
the Approximations of Exponentials of Logarithms (1967)
/ An important extension of Hilbert's approach to the
study of transcendence / 372 \\
\\
43. Eves. In Mathematical Circles; A Selection of
Mathematical Stories and Anecdotes (excerpt) (1969) / A
collection of mathematical stories and anecdotes about
$\pi$ / 400 \\
\\
44. Eves. Mathematical Circles Revisited; A Second
Collection of Mathematical Stories and Anecdotes
(excerpt) (1971) / A further collection of mathematical
stories and anecdotes about $\pi$ / 402 \\
\\
45. Todd. The Lemniscate Constants (1975) / A unifying
account of some of the methods used for computing the
lemniscate constants / 412 \\
\\
46. Salamin. Computation of r Using
Arithmetic-Geometric Mean (1976) / The first
quadratically converging algorithm for $\pi$ based on
Gauss's AGM and on Legendre's relation for elliptic
integrals / 418 \\
\\
47. Brent. Fast Multiple-Precision Evaluation of
Elementary Functions (1976) / ``This paper contains the
`Gauss-Legendre' method and some different algorithms
for log and exp (using Landen transformations).'' / 424
\\
\\
48. Beukers. A Note on the Irrationality of $\zeta(2)$
and $\zetq(3)$ (1979) / A short and elegant recasting
of Ap{\'e}ry's proof of the irrationality of $\zeta(3)$
(and $\zeta(2)$) / 434 \\
\\
49. van der Poorten. A Proof that Euler Missed \ldots{}
Ap{\'e}ry's Proof of the Irrationality of $\zeta(3)$
(1979) / An illuminating account of Ap{\'e}ry's
astonishing proof of the irrationality of $\zeta(3)$ /
439 \\
\\
50. Brent and McMillan. Some New Algorithms for
High-Precision Computation of Euler's Constant (1980) /
Several new algorithms for high precision calculation
of Euler's constant, including one which was used to
compute 30,100 decimal places / 448 \\
\\
51. Apostol. A Proof that Euler Missed: Evaluating
$\zeta(2)$ the Easy Way (1983) / This note shows that
one of the double integrals considered by Beukers ([48]
in the table of contents) can be used to establish
directly that $\zeta(2) = \pi / 6$ / 456 \\
\\
52. O'Shaughnessy. Putting God Back in Math (1983) / An
article about the Institute of Pi Research, an
organization that ``pokes fun at creationists by
pointing out that even the Bible makes mistakes.'' /
458 \\
\\
53. Stern. A Remarkable Approximation to $\pi$ (1985) /
Justification of the value of $\pi$ in the Bible
through numerological interpretations / 460 \\
\\
54. Newman and Shanks. On a Sequence Arising in Series
for $\pi$ (1984) / More connections between $\pi$ and
modular equations / 462 \\
\\
55. Cox. The Arithmetic-Geometric Mean of Gauss (1984)
/ An extensive study of the complex analytic properties
of the AGM / 481 \\
\\
56. Borwein and Borwein. The Arithmetic-Geometric Mean
and Fast Computation of Elementary Functions (1984) /
The relationship between the AGM iteration and fast
computation of elementary functions (one of the
by-products is an algorithm for $\pi$) / 537 \\
\\
57. Newman. A Simplified Version of the Fast Algorithms
of Brent and Salamin (1984) / Elementary algorithms for
evaluating $e^x$ and $\pi$ using the Gauss AGM without
explicit elliptic function theory / 553 \\
\\
58. Wagon. Is Pi Normal? (1985) / A discussion of the
conjecture that $\pi$ has randomly distributed digits /
557 \\
\\
59. Keith. Circle Digits: A Self-Referential Story
(1986) / A mnemonic for the first 402 decimal places of
$\pi$ / 560 \\
\\
60. Bailey. The Computation of $\pi$ to 29,360,000
Decimal Digits Using Borweins' Quartically Convergent
Algorithm (1988) / The algorithms used, both for $\pi$
and for performing the required multiple-precision
arithmetic / 562 \\
\\
61. Kanada. Vectorization of Multiple-Precision
Arithmetic Program and 201,326,000 Decimal Digits of 1
Calculation (1988) / Details of the computation and
statistical tests of the first 200 million digits of
$\pi$ / 576 \\
\\
62. Borwein and Borwein. Ramanujan and Pi (1988) / This
article documents Ramanujan's life, his ingenious
approach to calculating $\pi$, and how his approach is
now incorporated into modern computer algorithms / 588
\\
\\
63. Chudnovsky and Chudnovsky. Approximations and
Complex Multiplication According to Ramanujan (1988) /
This excerpt describes ``Ramanujan's original quadratic
period--quasiperiod relations for elliptic curves with
complex multiplication and their applications to
representations of fractions of $\pi$ and other
logarithms in terms of rapidly convergent nearly
integral (hypergeometric) series.'' / 596 \\
\\
64. Borwein, Borwein and Bailey. Ramanujan, Modular
Equations, and Approximations to Pi or How to Compute
One Billion Digits of Pi (1989) / An exposition of the
computation of $\pi$ using mathematics rooted in
Ramanujan's work / 623 \\
\\
65. Borwein, Borwein and Dilcher. Pi, Euler Numbers,
and Asymptotic Expansions (1989) / An explanation as to
why the slowly convergent Gregory series for $\pi$,
truncated at 500,000 terms, gives $\pi$ to 40 places
with only the 6th, 17th, 18th, and 29th places being
incorrect / 642 \\
\\
66. Beukers, B{\'e}zivin, and Robba. An Alternative
Proof of the Lindemann--Weierstrass Theorem (1990) /
The Lindemann--Weierstrass theorem as a by-product of a
criterion for rationality of solutions of differential
equations / 649 \\
\\
67. Webster. The Tail of Pi (1991) / Various anecdotes
about $\pi$ from the 14th annual IMO Lecture to the
Royal Society / 654 \\
\\
68. Eco. An excerpt from Foucault's Pendulum (1993) /
``The unnumbered perfection of the circle itself.'' /
658 \\
\\
69. Keith. Pi Mnemonics and the Art of Constrained
Writing (1996) / A mnemonic for $\pi$ based on Edgar
Allen Poe's poem ``The Raven.'' / 659 \\
\\
70. Bailey, Borwein, and Plouffe. On the Rapid
Computation of Various Polylogarithmic Constants (1996)
/ A fast method for computing individual digits of
$\pi$ in base 2 / 663 \\
Appendix I --- On the Early History of Pi / 677 \\
\\
Appendix II --- A Computational Chronology of Pi / 683
\\
\\
Appendix III --- Selected Formulae for Pi / 686 \\
\\
Bibliography / 690 \\
\\
Credits / 697 \\
\\
Index / 701",
}
@Book{Berggren:2000:PSB,
editor = "Lennart Berggren and Jonathan Borwein and Peter
Borwein",
booktitle = "Pi: a source book",
title = "Pi: a source book",
publisher = pub-SV,
address = pub-SV:adr,
edition = "Second",
pages = "xx + 736",
year = "2000",
DOI = "https://doi.org/10.1007/978-1-4757-3240-5",
ISBN = "0-387-98946-3 (hardcover)",
ISBN-13 = "978-0-387-98946-4 (hardcover)",
LCCN = "QA484 .P5 2000",
MRclass = "11-00 (01A05 01A75 11-03)",
MRnumber = "1746004",
bibdate = "Wed Aug 10 11:09:47 2016",
bibsource = "https://www.math.utah.edu/pub/bibnet/authors/b/borwein-jonathan-m.bib;
https://www.math.utah.edu/pub/tex/bib/agm.bib;
https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
https://www.math.utah.edu/pub/tex/bib/master.bib;
https://www.math.utah.edu/pub/tex/bib/mathcw.bib;
https://www.math.utah.edu/pub/tex/bib/pi.bib",
acknowledgement = ack-nhfb,
author-dates = "Jonathan Michael Borwein (20 May 1951--2 August
2016)",
libnote = "Not yet in my library.",
ORCID-numbers = "Borwein, Jonathan/0000-0002-1263-0646",
subject = "Pi (mathematical constant)",
tableofcontents = "Preface / v \\
\\
Preface to the Second Edition / viii \\
Acknowledgments / ix \\
\\
Introduction / xvii \\
\\
1. The Rhind Mathematical Papyrus-Problem 50 ($\approx$
1650 B.C.) / A problem dealing with the area of a round
field of given diameter / 1 \\
\\
2. Engels. Quadrature of the Circle in Ancient Egypt
(1977) / A conjectural explanation of how the
mathematicians of ancient Egypt approximated the area
of a circle / 3 \\
\\
3. Archimedes. Measurement of a Circle ($\approx$ 250
BC) / The seminal work in which Archimedes presents the
first true algorithm for $\pi$ / 7 \\
\\
4. Phillips. Archimedes the Numerical Analyst (1981) /
A summary of Archimedes' work on the computation of
$\pi$ using modern notation / 15 \\
\\
5. Lam and Ang. Circle Measurements in Ancient China
(1986) / This paper discusses and contains a
translation of Liu Hui's (3rd century) method for
evaluating $\pi$ and also examines values for $\pi$
given by Zu Chongzhi (429--500) / 20 \\
\\
6. The Ban{\=u} M{\=u}s{\=a}: The Measurement of Plane
and Solid Figures ($\approx$ 850) / This extract gives
an explicit statement and proof that the ratio of the
circumference to the diameter is constant / 36 \\
\\
7. M{\=a}dhava. The Power Series for Arctan and Pi
($\approx$ 1400) / These theorems by a fifteenth
century Indian mathematician give Gregory's series for
arctan with remainder terms and Leibniz's series for
$\pi$ / 45 \\
\\
8. Hope-Jones. Ludolph (or Ludolff or Lucius) van
Ceulen (1938) / Correspondence about van Ceulen's
tombstone in reference to it containing some digits of
$\pi$ / 51 \\
\\
9. Vi{\'e}te. Variorum de Rebus Mathematicis Reponsorum
Liber VII (1593) / Two excerpts. One containing the
first infinite expression of $\pi$, obtained by
relating the area of a regular $2n$-gon to that of a
regular $n$-gon / 53 \\
\\
10. Wallis. Computation of $\pi$ by Successive
Interpolations (1655) / How Wallis derived the infinite
product for $\pi$ that bears his name / 68 \\
\\
11. Wallis. Arithmetica Infinitorum (1655) / An excerpt
including Prop. 189, 191 and an alternate form of the
result that gives Wm. Brounker's continued fraction
expression for $4/\pi$ / 78 \\
\\
12. Huygens. De Circuli Magnitudine Inventa (1724) /
Huygens's proof of W. Snell's discovery of improvements
in Archimedes' method of estimating the lengths of
circular arcs / 81 \\
\\
13. Gregory. Correspondence with John Collins (1671) /
A letter to Collins in which he gives his series for
arctangent, carried to the ninth power. / 87 \\
\\
14. Roy. The Discovery of the Series Formula for $\pi$
by Leibniz, Gregory, and Nilakantha (1990) / A
discussion of the discovery of the series $\pi/4 = 1 -
1/3 + 1/5, \cdots{}$ / 92 \\
\\
15. Jones. The First Use of $\pi$ for the Circle Ratio
(1706) / An excerpt from Jones' book, the Synopsis
Palmariorum Matheseos: or, a New Introduction to the
Mathematics, London, 1706 / 108 \\
\\
16. Newton. Of the Method of Fluxions and Infinite
Series (1737) / An excerpt giving Newton's calculation
of $\pi$ to 16 decimal places / 110 \\
\\
17. Euler. Chapter 10 of Introduction to Analysis of
the Infinite (On the Use of the Discovered Fractions to
Sum Infinite Series) (1748) / This includes many of
Euler's infinite series for $\pi$ and powers of $\pi$ /
112 \\
\\
18. Lambert. M{\'e}moire Sur Quelques
Propri{\'e}t{\'e}s Remarquables Des Quantit{\'e}s
Transcendentes Circulaires et Logarithmiques (1761) /
An excerpt from Lambert's original proof of the
irrationality of $\pi$ / 129 \\
\\
19. Lambert. Irrationality of $\pi$ (1969) / A
translation and Struik's discussion of Lambert's proof
of the irrationality of $\pi$ / 141 \\
\\
20. Shanks. Contributions to Mathematics Comprising
Chiefly of the Rectification of the Circle to 607
Places of Decimals (1853) / Pages from Shank's report
of his monumental hand calculation of $\pi$ / 147 \\
\\
21. Hermite. Sur La Fonction Exponentielle (1873) / The
first proof of the transcendence of $e$ / 162 \\
\\
22. Lindemann. Ueber die Zahl $\pi$ (1882) / The first
proof of the transcendence of $\pi$ / 194 \\
\\
23. Weierstrass. Zu Lindemann's Abhandlung ``Uber die
Ludolphsche Zahl'' (1885) / Weierstrass' proof of the
transcendence of $\pi$ / 207 \\
\\
24. Hilbert. Ueber die Trancendenz der Zahlen $e$ und
$\pi$ (1893) / Hilbert's short and elegant
simplification of the transcendence proofs for $e$ and
$\pi$ / 226 \\
\\
25. Goodwin. Quadrature of the Circle (1894) / The
dubious origin of the attempted legislation of the
value of $\pi$ in Indiana / 230 \\
\\
26. Edington. House Bill No. 246, Indiana State
Legislature, 1897 (1935) / A summary of the action
taken by the Indiana State Legislature to fix the value
of $\pi$ (including a copy of the actual bill that was
proposed) / 231 \\
\\
27. Singmaster. The Legal Values of Pi (1985) / A
history of the attempt by Indiana to legislate the
value of $\pi$ / 236 \\
\\
28. Ramanujan. Squaring the Circle (1913) / A geometric
approximation to $\pi$ / 240 \\
\\
29. Ramanujan. Modular Equations and Approximations to
$\pi$ (1914) / Ramanujan's seminal paper on $\pi$ that
includes a number of striking series and algebraic
approximations / 241 \\
\\
30. Watson. The Marquis and the Land Agent: A Tale of
the Eighteenth Century (1933) / A Presidential address
to the Mathematical Association in which the author
gives an account of ``some of the elementary work on
arcs and ellipses and other curves which led up to the
idea of inverting an elliptic integral, and so laying
the foundations of elliptic functions and doubly
periodic functions generally.'' / 258 \\
\\
31. Ballantine. The Best (?) Formula for Computing
$\pi$ to a Thousand Places (1939) / An early attempt to
orchestrate the calculation of $\pi$ more cleverly /
271 \\
\\
32. Birch. An Algorithm for Construction of Arctangent
Relations (1946) / The object of this note is to
express $\pi / 4 $ as a sum of arctan relations in
powers of 10 / 274 \\
\\
33. Niven. A Simple Proof that $\pi$ Is Irrational
(1947) / A very concise proof of the irrationality of
$\pi$ / 276 \\
\\
34. Reitwiesner. An ENIAC Determination of $\pi$ and
$e$ to 2000 Decimal Places (1950) / One of the first
computer-based computations / 277 \\
\\
35. Schepler. The Chronology of Pi (1950) / A fairly
reliable outline of the history of $\pi$ from 3000 BC
to 1949 / 282 \\
\\
36. Mahler. On the Approximation of $\pi$ (1953) /
``The aim of this paper is to determine an explicit
lower bound free of unknown constants for the distance
of $\pi$ from a given rational or algebraic number'' /
306 \\
\\
37. Wrench, Jr. The Evolution of Extended Decimal
Approximations to $\pi$ (1960) / A history of the
calculation of the digits of $\pi$ to 1960 \\
\\
38. Shanks and Wrench, Jr. Calculation of $\pi$ to
100,000 Decimals (1962) / A landmark computation of
$\pi$ to more than 100,000 places / 326 \\
\\
39. Sweeny. On the Computation of Euler's Constant
(1963) / The computation of Euler's constant to 3566
decimal places / 350 \\
\\
40. Baker. Approximations to the Logarithms of Certain
Rational Numbers (1964) / The main purpose of this deep
and fundamental paper is to ``deduce results concerning
the accuracy with which the natural logarithms of
certain rational numbers may be approximated by
rational numbers, or, more generally, by algebraic
numbers of bounded degree.'' / 359 \\
\\
41. Adams. Asymptotic Diophantine Approximations to $E$
(1966) / An asymptotic estimate for the rational
approximation to $e$ which disproves the conjecture
that $e$ behaves like almost all numbers in this
respect / 368 \\
\\
42. Mahler. Applications of Some Formulae by Hermite to
the Approximations of Exponentials of Logarithms (1967)
/ An important extension of Hilbert's approach to the
study of transcendence / 372 \\
\\
43. Eves. In Mathematical Circles; A Selection of
Mathematical Stories and Anecdotes (excerpt) (1969) / A
collection of mathematical stories and anecdotes about
$\pi$ / 400 \\
\\
44. Eves. Mathematical Circles Revisited; A Second
Collection of Mathematical Stories and Anecdotes
(excerpt) (1971) / A further collection of mathematical
stories and anecdotes about $\pi$ / 402 \\
\\
45. Todd. The Lemniscate Constants (1975) / A unifying
account of some of the methods used for computing the
lemniscate constants / 412 \\
\\
46. Salamin. Computation of r Using
Arithmetic-Geometric Mean (1976) / The first
quadratically converging algorithm for $\pi$ based on
Gauss's AGM and on Legendre's relation for elliptic
integrals / 418 \\
\\
47. Brent. Fast Multiple-Precision Evaluation of
Elementary Functions (1976) / ``This paper contains the
`Gauss-Legendre' method and some different algorithms
for log and exp (using Landen transformations).'' / 424
\\
\\
48. Beukers. A Note on the Irrationality of $\zeta(2)$
and $\zetq(3)$ (1979) / A short and elegant recasting
of Ap{\'e}ry's proof of the irrationality of $\zeta(3)$
(and $\zeta(2)$) / 434 \\
\\
49. van der Poorten. A Proof that Euler Missed \ldots{}
Ap{\'e}ry's Proof of the Irrationality of $\zeta(3)$
(1979) / An illuminating account of Ap{\'e}ry's
astonishing proof of the irrationality of $\zeta(3)$ /
439 \\
\\
50. Brent and McMillan. Some New Algorithms for
High-Precision Computation of Euler's Constant (1980) /
Several new algorithms for high precision calculation
of Euler's constant, including one which was used to
compute 30,100 decimal places / 448 \\
\\
51. Apostol. A Proof that Euler Missed: Evaluating
$\zeta(2)$ the Easy Way (1983) / This note shows that
one of the double integrals considered by Beukers ([48]
in the table of contents) can be used to establish
directly that $\zeta(2) = \pi / 6$ / 456 \\
\\
52. O'Shaughnessy. Putting God Back in Math (1983) / An
article about the Institute of Pi Research, an
organization that ``pokes fun at creationists by
pointing out that even the Bible makes mistakes.'' /
458 \\
\\
53. Stern. A Remarkable Approximation to $\pi$ (1985) /
Justification of the value of $\pi$ in the Bible
through numerological interpretations / 460 \\
\\
54. Newman and Shanks. On a Sequence Arising in Series
for $\pi$ (1984) / More connections between $\pi$ and
modular equations / 462 \\
\\
55. Cox. The Arithmetic-Geometric Mean of Gauss (1984)
/ An extensive study of the complex analytic properties
of the AGM / 481 \\
\\
56. Borwein and Borwein. The Arithmetic-Geometric Mean
and Fast Computation of Elementary Functions (1984) /
The relationship between the AGM iteration and fast
computation of elementary functions (one of the
by-products is an algorithm for $\pi$) / 537 \\
\\
57. Newman. A Simplified Version of the Fast Algorithms
of Brent and Salamin (1984) / Elementary algorithms for
evaluating $e^x$ and $\pi$ using the Gauss AGM without
explicit elliptic function theory / 553 \\
\\
58. Wagon. Is Pi Normal? (1985) / A discussion of the
conjecture that $\pi$ has randomly distributed digits /
557 \\
\\
59. Keith. Circle Digits: A Self-Referential Story
(1986) / A mnemonic for the first 402 decimal places of
$\pi$ / 560 \\
\\
60. Bailey. The Computation of $\pi$ to 29,360,000
Decimal Digits Using Borweins' Quartically Convergent
Algorithm (1988) / The algorithms used, both for $\pi$
and for performing the required multiple-precision
arithmetic / 562 \\
\\
61. Kanada. Vectorization of Multiple-Precision
Arithmetic Program and 201,326,000 Decimal Digits of 1
Calculation (1988) / Details of the computation and
statistical tests of the first 200 million digits of
$\pi$ / 576 \\
\\
62. Borwein and Borwein. Ramanujan and Pi (1988) / This
article documents Ramanujan's life, his ingenious
approach to calculating $\pi$, and how his approach is
now incorporated into modern computer algorithms / 588
\\
\\
63. Chudnovsky and Chudnovsky. Approximations and
Complex Multiplication According to Ramanujan (1988) /
This excerpt describes ``Ramanujan's original quadratic
period--quasiperiod relations for elliptic curves with
complex multiplication and their applications to
representations of fractions of $\pi$ and other
logarithms in terms of rapidly convergent nearly
integral (hypergeometric) series.'' / 596 \\
\\
64. Borwein, Borwein and Bailey. Ramanujan, Modular
Equations, and Approximations to Pi or How to Compute
One Billion Digits of Pi (1989) / An exposition of the
computation of $\pi$ using mathematics rooted in
Ramanujan's work / 623 \\
\\
65. Borwein, Borwein and Dilcher. Pi, Euler Numbers,
and Asymptotic Expansions (1989) / An explanation as to
why the slowly convergent Gregory series for $\pi$,
truncated at 500,000 terms, gives $\pi$ to 40 places
with only the 6th, 17th, 18th, and 29th places being
incorrect / 642 \\
\\
66. Beukers, B{\'e}zivin, and Robba. An Alternative
Proof of the Lindemann--Weierstrass Theorem (1990) /
The Lindemann--Weierstrass theorem as a by-product of a
criterion for rationality of solutions of differential
equations / 649 \\
\\
67. Webster. The Tail of Pi (1991) / Various anecdotes
about $\pi$ from the 14th annual IMO Lecture to the
Royal Society / 654 \\
\\
68. Eco. An excerpt from Foucault's Pendulum (1993) /
``The unnumbered perfection of the circle itself.'' /
658 \\
\\
69. Keith. Pi Mnemonics and the Art of Constrained
Writing (1996) / A mnemonic for $\pi$ based on Edgar
Allen Poe's poem ``The Raven.'' / 659 \\
\\
70. Bailey, Borwein, and Plouffe. On the Rapid
Computation of Various Polylogarithmic Constants (1996)
/ A fast method for computing individual digits of
$\pi$ in base 2 / 663 \\
Appendix I --- On the Early History of Pi / 677 \\
\\
Appendix II --- A Computational Chronology of Pi / 683
\\
\\
Appendix III --- Selected Formulae for Pi / 686 \\
\\
Appendix IV --- Translations of Vi{\`e}te and Huygens /
690 \\
Bibliography / 711 \\
\\
Credits / 717 \\
\\
Index / 721",
}
@Book{Berggren:2004:PSB,
editor = "Lennart Berggren and Jonathan Borwein and Peter
Borwein",
booktitle = "Pi: a source book",
title = "Pi: a source book",
publisher = pub-SV,
address = pub-SV:adr,
edition = "Third",
pages = "xx + 797",
year = "2004",
DOI = "https://doi.org/10.1007/978-1-4757-4217-6",
ISBN = "0-387-20571-3",
ISBN-13 = "978-0-387-20571-7",
MRclass = "11-00 (01A05 01A75 11-03)",
MRnumber = "2065455",
MRreviewer = "F. Beukers",
bibdate = "Wed Aug 10 11:09:47 2016",
bibsource = "https://www.math.utah.edu/pub/bibnet/authors/b/borwein-jonathan-m.bib;
https://www.math.utah.edu/pub/tex/bib/agm.bib;
https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
https://www.math.utah.edu/pub/tex/bib/pi.bib",
acknowledgement = ack-nhfb,
author-dates = "Jonathan Michael Borwein (20 May 1951--2 August
2016)",
ORCID-numbers = "Borwein, Jonathan/0000-0002-1263-0646",
remark = "CECM Preprint 2003:210.",
tableofcontents = "Preface to the Third Edition / v \\
Preface to the Second Edition / vi \\
Preface / vii \\
Acknowledgments / x \\
Introduction / xvii \\
1. The Rhind Mathematical Papyrus --- Problem 50
($\approx$1650 B.C.) / A problem dealing with the area
of a round field of given diameter / 1 \\
2. Engels / Quadrature of the Circle in Ancient Egypt
(1977) / A conjectural explanation of how the
mathematicians of ancient Egypt approximated the area
of a circle / 3 \\
3. Archimedes / Measurement of a Circle --- (-250 B.C.)
/ The seminal work in which Archimedes presents the
first true algorithm for $ \pi $ / 7 \\
4. Phillips / Archimedes the Numerical --- Analyst
(1981) / A summary of Archimedes' work on the
computation of $ \pi $ using modem notation / 15 \\
5. Lam and Ang / Circle Measurements in Ancient China
(1986) / This paper discusses and contains a
translation of Liu Hui's (3rd century) method for
evaluating $ \pi $ and also examines values for $ \pi $
given by Zu Chongzhi (429--500) / 20 \\
6. The Ban{\=u} M{\=u}s{\=a}: The Measurement of Plane
and Solid Figures (--850) / This extract gives an
explicit statement and proof that the ratio of the
circumference to the diameter is constant / 36 \\
7. M{\=a}dhava / The Power Series for Arctan and Pi
(-1400) / These theorems by a fifteenth century Indian
mathematician give Gregory's series for arctan with
remainder terms and Leibniz's series for $ \pi $ / 45
\\
8. Hope-Jones / Ludolph (or Ludolff or Lucius) van
Ceulen (1938) / Correspondence about van Ceulen's
tombstone in reference to it containing some digits of
$ \pi $ / 51 \\
9. Vi{\`e}te / \booktitle{Variorum de Rebus
Mathematicis Reponsorum Liber VII} (1593) / Two
excerpts. One containing the first infinite expression
of $ \pi $, obtained by relating the area of a regular
$2n$-gon to that of a regular $n$-gon / 53 \\
10. Wallis. Computation of $ \pi $ by Successive
Interpolations (1655) / How Wallis derived the infinite
product for $ \pi $ that bears his name / 68 \\
11. Wallis / \booktitle{Arithmetica Infinitorum} (1655)
/ An excerpt including Prop. 189, 191 and an alternate
form of the result that gives Wm. Brounker's continued
fraction expression for $ 4 / \pi$ / ?? \\
12. Huygens / \booktitle{De Circuli Magnitudine
Inventa} (1654) / Huygens's demonstration of how to
triple the number of correct decimals over those in
Archimedes' estimate of $ \pi $ / 81 13. Gregory /
Correspondence with John Collins (1671) / A letter to
Collins in which he gives his series for arctangent,
carried to the ninth power / 87 \\
14. Roy / The Discovery of the Series Formula for $ \pi
$ by Leibniz, Gregory, and Nilakantha (1990) / A
discussion of the discovery of the series $ \pi / 4 = 1
- 1/3 + 1/5 - \cdots{} $ / 92 \\
15. Jones / The First Use of $ \pi $ for the Circle
Ratio (1706) / An excerpt from Jones' book, the
\booktitle{Synopsis Palmariorum Matheseos: or, a New
Introduction to the Mathematics}, London, 1706 / 108
\\
16. Newton / Of the Method of Fluxions and Infinite
Series (1737) / An excerpt giving Newton's calculation
of $ \pi $ to 16 decimal places / 110 \\
17. Euler / Chapter 10 of \booktitle{Introduction to
Analysis of the Infinite (On the Use of the Discovered
Fractions to Sum Infinite Series)} (1748) / This
includes many of Euler's infinite series for $ \pi $
and powers of $ \pi $ / 112 \\
18. Lambert / \booktitle{M{\'e}moire Sur Quelques
Propri{\'e}t{\'e}s Remarquables Des Quantit{\'e}s
Transcendentes Circulaires et Logarithmiques} (1761) /
An excerpt from Lambert's original proof of the
irrationality of $ \pi $ / 129 19. Lambert /
Irrationality of $ \pi $ (1969) / A translation and
Struik's discussion of Lambert's proof of the
irrationality of $ \pi $ / 141 \\
20. Shanks / Contributions to Mathematics Comprising
Chiefly of the Rectification of the Circle to 607
Places of Decimals (1853) / Pages from Shanks's report
of his monumental hand calculation of $ \pi $ / 147 \\
21. Hermite / \booktitle{Sur La Fonction Exponentielle}
(1873) / The first proof of the transcendence of $ e $
/ 162 \\
22. Lindemann / \booktitle{Ueber die Zahl $ \pi $}
(1882) / The first proof of the transcendence of $ \pi
$ / 194 23. Weierstrass / \booktitle{Zu Lindemann's
Abhandlung ``{\"U}ber die Ludolphsche Zahl''} (1885) /
Weierstrass' proof of the transcendence of $ \pi $ /
207 24. Hilbert / \booktitle{Ueber die Transzendenz der
Zahlen $ e $ und $ \pi $} (1893) / Hilbert's short and
elegant simplification of the transcendence proofs for
$ e $ and $ \pi $ / 226 25. Goodwin / Quadrature of the
Circle (1894) / The dubious origin of the attempted
legislation of the value of $ \pi $ in Indiana / 230
\\
26. Edington / House Bill No. 246, Indiana State
Legislature, 1897 (1935) / A summary of the action
taken by the Indiana State Legislature to fix the value
of $ \pi $ (including a copy of the actual bill that
was proposed) / 231 \\
27. Singmaster / The Legal Values of Pi (1985) / A
history of the attempt by Indiana to legislate the
value of $ \pi $ / 236 \\
28. Ramanujan / Squaring the Circle (1913) / A
geometric approximation to $ \pi $ / 240 \\
29. Ramanujan / Modular Equations and Approximations to
$ \pi $ (1914) / Ramanujan's seminal paper on pi that
includes a number of striking series and algebraic
approximations / 241 \\
30. Watson / The Marquis and the Land Agent: A Tale of
the Eighteenth Century (1933) / A Presidential address
to the Mathematical Association in which the author
gives an account of ``some of the elementary work on
arcs and ellipses and other curves which led up to the
idea of inverting an elliptic integral, and so laying
the foundations of elliptic functions and doubly
periodic functions generally.'' / ?? \\
31. Ballantine / The Best (?) Formula for Computing $
\pi $ to a Thousand Places (1939) / An early attempt to
orchestrate the calculation of $ \pi $ more cleverly /
271 \\
32. Birch / An Algorithm for Construction of Arctangent
Relations (1946) / The object of this note is to
express $ \pi / 4$ as a sum of arctan relations in
powers of 10 / 274 \\
33. Niven / A Simple Proof that $ \pi $ is Irrational
(1947) / A very concise proof of the irrationality of $
\pi $ / 276 \\
34. Reitwiesner / An ENIAC Determination of $ \pi $ and
$ e $ to 2000 Decimal Places (1950) / One of the first
computer-based computations / 277 \\
35. Schepler / The Chronology of Pi (1950) / A fairly
reliable outline of the history of $ \pi $ from 3000
B.C. to 1949 / 282 \\
36. Mahler / On the Approximation of $ \pi $ (1953) /
``The aim of this paper is to determine an explicit
lower bound free of unknown constants for the distance
of $ \pi $ from a given rational or algebraic number.''
/ 306 \\
37. Wrench, Jr. / The Evolution of Extended Decimal
Approximations to $ \pi $ (1960) / A history of the
calculation of the digits of $ \pi $ to 1960 / 319 \\
38. Shanks and Wrench, Jr. / Calculation of $ \pi $ to
100,000 Decimals (1962) / A landmark computation of $
\pi $ to more than 100,000 places / 326 39. Sweeny / On
the Computation of Euler's Constant (1963) / The
computation of Euler's constant to 3566 decimal places
/ 350 40. Baker / Approximations to the Logarithms of
Certain Rational Numbers (1964) / The main purpose of
this deep and fundamental paper is to ``deduce results
concerning the accuracy with which the natural
logarithms of certain rational numbers may be
approximated by rational numbers, or, more generally,
by algebraic numbers of bounded degree.'' / 359 \\
41. Adams / Asymptotic Diophantine Approximations to e
(1966) / An asymptotic estimate for the rational
approximation to $ e $ which disproves the conjecture
that $ e $ behaves like almost all numbers in this
respect / 368 \\
42. Mahler / Applications of Some Formulae by Hermite
to the Approximations of Exponentials of Logarithms
(1967) / An important extension of Hilbert's approach
to the study of transcendence / 372 43. Eves / In
Mathematical Circles; A Selection of Mathematical
Stories and Anecdotes (excerpt) (1969) / A collection
of mathematical stories and anecdotes about $ \pi $ /
456 \\
44. Eves / Mathematical Circles Revisited; A Second
Collection of Mathematical Stories and Anecdotes
(excerpt) (1971) / A further collection of mathematical
stories and anecdotes about $ \pi $ / 402 45. Todd /
The Lemniscate Constants (1975) / A unifying account of
some of the methods used for computing the lemniscate
constants / 412 \\
46. Salamin / Computation of $ \pi $ Using
Arithmetic--Geometric Mean (1976) / The first
quadratically converging algorithm for $ \pi $ based on
Gauss's AGM and on Legendre's relation for elliptic
integrals / 418 \\
47. Brent / Fast Multiple-Precision Evaluation of
Elementary Functions (1976) / ``This paper contains the
`Gauss--Legendre' method and some different algorithms
for $\log$ and $\exp$ (using Landen transformations).''
/ 424 \\
48. Beukers / A Note on the Irrationality of $ \zeta(2)
$ and $ \zeta(3) $ (1979) / A short and elegant
recasting of Apery's proof of the irrationality of
$\zeta(3)$ (and $\zeta(2)$) / 434 \\
49. van der Poorten / A Proof that Euler Missed
\ldots{} Apery's Proof of the Irrationality of $\zeta
(3)$ (1979) / An illuminating account of Apery's
astonishing proof of the irrationality of $\zeta (3)$ /
439 \\
50. Brent and McMillan / Some New Algorithms for
High-Precision Computation of Euler's Constant (1980) /
Several new algorithms for high-precision calculation
of Euler's constant, including one which was used to
compute 30,100 decimal places / 448 \\
51. Apostol / A Proof that Euler Missed: Evaluating
$\zeta(2)$ the Easy Way (1983) / This note shows that
one of the double integrals considered by Beukers ([48]
in the table of contents) can be used to establish
directly that $\zeta(2) = \pi^2 / 6$ / 456 \\
52. O'Shaughnessy / Putting God Back in Math (1983) /
An article about the Institute of Pi Research, an
organization that ``pokes fun at creationists by
pointing out that even the Bible makes mistakes.'' /
458 \\
53. Stern / A Remarkable Approximation to $ \pi $
(1985) / Justification of the value of $ \pi $ in the
Bible through numerological interpretations / 460 \\
54. Newman and Shanks / On a Sequence Arising in Series
for $ \pi $ (1984) / More connections between $ \pi $
and modular equations / 462 \\
55. Cox / The Arithmetic--Geometric Mean of Gauss
(1984) / An extensive study of the complex analytic
properties of the AGM / 481 \\
56. Borwein and Borwein / The Arithmetic--Geometric
Mean and Fast Computation of Elementary Functions
(1984) / The relationship between the AGM iteration and
fast computation of elementary functions (one of the
by-products is an algorithm for $ \pi $) / 537 57.
Newman / A Simplified Version of the Fast Algorithms of
Brent and Salamin (1984) / Elementary algorithms for
evaluating $ e^x $ and $ \pi $ using the Gauss AGM
without explicit elliptic function theory / 553 \\
58. Wagon / Is Pi Normal? (1985) / A discussion of the
conjecture that $ \pi $ has randomly distributed digits
/ 557 \\
59. Keith / Circle Digits: A Self-Referential Story
(1986) / A mnemonic for the first 402 decimal places of
$ \pi $ / 560 \\
60. Bailey / The Computation of $ \pi $ to 29,360,000
Decimal Digits Using Borwein's Quartically Convergent
Algorithm (1988) / The algorithms used, both for $ \pi
$ and for performing the required multiple-precision
arithmetic / 562 \\
61. Kanada / Vectorization of Multiple-Precision
Arithmetic Program and 201,326,000 Decimal Digits of $
\pi $ Calculation (1988) / Details of the computation
and statistical tests of the first 200 million digits
of $ \pi $ / 576 \\
62. Borwein and Borwein / Ramanujan and Pi (1988) /
This article documents Ramanujan's life, his ingenious
approach to calculating $ \pi $, and how his approach
is now incorporated into modern computer algorithms /
588 \\
63. Chudnovsky and Chudnovsky / Approximations and
Complex Multiplication According to Ramanujan (1988) /
This excerpt describes ``Ramanujan's original quadratic
period--quasiperiod relations for elliptic curves with
complex multiplication and their applications to
representations of fractions of $ \pi $ and other
logarithms in terms of rapidly convergent nearly
integral (hypergeometric) series.'' / 596 \\
64. Borwein, Borwein and Bailey / Ramanujan, Modular
Equations, and Approximations to Pi or How to Compute
One Billion Digits of Pi (1989) / An exposition of the
computation of $ \pi $ using mathematics rooted in
Ramanujan's work / 623 \\
65. Borwein, Borwein and Dilcher / Pi, Euler Numbers,
and Asymptotic Expansions (1989) / An explanation as to
why the slowly convergent Gregory series for $ \pi $,
truncated at 500,000 terms, gives $ \pi $ to 40 places
with only the 6th, 17th, 18th, and 29th places being
incorrect / 642 \\
66. Beukers, Bezivin, and Robba / An Alternative Proof
of the Lindemann--Weierstrass Theorem (1990) / The
Lindemann--Weierstrass theorem as a by-product of a
criterion for rationality of solutions of differential
equations / 649 \\
67. Webster / The Tale of Pi (1991) / Various anecdotes
about $ \pi $ from the 14th annual IMO Lecture to the
Royal Society / 654 \\
68. Eco / An excerpt from Foucault's Pendulum (1993) /
``The unnumbered perfection of the circle itself.'' /
658 \\
69. Keith / Pi Mnemonics and the Art of Constrained
Writing (1996) / A mnemonic for $ \pi $ based on Edgar
Allen Poe's poem ``The Raven.'' / 659 \\
70. Bailey, Borwein, and Plouffe / On the Rapid
Computation of Various Polylogarithmic Constants (1997)
/ A fast method for computing individual digits of $
\pi $ in base 2 / 663 \\
Appendix I --- On the Early History of Pi / 677 \\
Appendix II --- A Computational Chronology of Pi / 683
\\
Appendix III --- Selected Formulae for Pi / 686 \\
Appendix IV --- Translations of Viele and Huygens / 690
\\
Bibliography / 710 \\
Credits / 717 \\
A Pamphlet on Pi / 721 \\
Contents / 723 \\
1. Pi and Its Friends / 725 \\
2. Normality of Numbers / 741 \\
3. Historia Cyclometrica / 753 \\
4. Demotica Cyclometrica / 771 \\
References / 779 \\
Index / 783",
}
@Book{Schumer:2004:MJ,
author = "Peter D. Schumer",
booktitle = "Mathematical journeys",
title = "Mathematical journeys",
publisher = pub-WI,
address = pub-WI:adr,
pages = "xi + 199",
year = "2004",
ISBN = "0-471-22066-3 (paperback)",
ISBN-13 = "978-0-471-22066-4 (paperback)",
LCCN = "QA93 .S38 2004",
bibdate = "Sat Sep 10 16:27:33 MDT 2016",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib;
z3950.loc.gov:7090/Voyager",
URL = "http://www.loc.gov/catdir/bios/wiley044/2003062040.html;
http://www.loc.gov/catdir/description/wiley041/2003062040.html;
http://www.loc.gov/catdir/toc/wiley041/2003062040.html",
acknowledgement = ack-nhfb,
author-dates = "1954--",
subject = "Mathematics; Popular works",
tableofcontents = "Preface / ix \\
Acknowledgments / xi \\
1. Let's Get Cooking: A Variety of Mathematical
Ingredients / 1 \\
2. The Green Chicken Contest / 13 \\
3. The Josephus Problem: Please Choose Me Last / 23 \\
4. Nim and Wythoff's Game: Or How to Get Others to Pay
Your Bar Bill / 31 \\
5. Mersenne Primes, Perfect Numbers, and Amicable Pairs
/ 41 \\
6. The Harmonic Series \ldots{} and Less / 49 \\
7. Fermat Primes, the Chinese Remainder Theorem, and
Lattice Points / 61 \\
8. Tic-Tac-Toe, Magic Squares, and Latin Squares / 71
\\
9. Mathematical Variations on Rolling Dice / 83 \\
10. Pizza Slicing, Map Coloring, Pointillism, and
Jack-in-the-Box / 91 \\
11. Episodes in the Calculation of Pi / 101 / \\
12. A Sextet of Scintillating Problems / 117 \\
13. Primality Testing Below a Quadrillion / 127 \\
14. Erd{\H{o}}s Number Zero / 139 \\
15. Choosing Stamps to Mail a Letter, Let Me Count the
Ways / 153 \\
16. Pascal Potpourri / 165 \\
Appendix: Comments and Solutions to Problems Worth
Considering / 177 \\
Bibliography / 193 \\
Index / 195",
}
@Book{Alladi:2013:RPW,
author = "Krishnaswami Alladi",
booktitle = "{Ramanujan}'s place in the world of mathematics:
essays providing a comparative study",
title = "{Ramanujan}'s place in the world of mathematics:
essays providing a comparative study",
publisher = "Springer",
address = "New Delhi, India",
pages = "xviii + 177",
year = "2013",
DOI = "https://doi.org/10.1007/978-81-322-0767-2",
ISBN = "81-322-0766-1 (print), 81-322-0767-X (electronic)",
ISBN-13 = "978-81-322-0766-5 (print), 978-81-322-0767-2
(electronic)",
LCCN = "QA29.R3 A65 2013",
bibdate = "Sat Sep 3 16:45:57 MDT 2016",
bibsource = "https://www.math.utah.edu/pub/tex/bib/pi.bib;
z3950.loc.gov:7090/Voyager",
abstract = "This book is a collection of articles, all by the
author, on the Indian mathematical genius Srinivasa
Ramanujan as well as on some of the greatest
mathematicians in history whose lives and works have
things in common with Ramanujan. It presents a unique
comparative study of Ramanujan's spectacular
discoveries and remarkable life with the monumental
contributions of various mathematical luminaries, some
of whom, like Ramanujan, overcame great difficulties in
life. Also, among the articles are reviews of three
important books on Ramanujan's mathematics and life. In
addition, some aspects of Ramanujan's contributions,
such as his remarkable formulae for the number $ \pi $,
his pathbreaking work in the theory of partitions, and
his fundamental observations on quadratic forms, are
discussed. Finally, the book describes various current
efforts to ensure that the legacy of Ramanujan will be
preserved and continue to thrive in the future. Thus
the book is an enlightening study of Ramanujan as a
mathematician and a human being.",
acknowledgement = ack-nhfb,
subject = "Ramanujan Aiyangar, Srinivasa; Mathematicians; India;
History; Biography",
subject-dates = "1887--1920",
tableofcontents = "Part 1: Ramanujan and other mathematical luminaries
\\
Ramanujan: An Estimation \\
Ramanujan: The Second Century \\
L. J. Rogers: A Contemporary of Ramanujan \\
P. A. MacMahon: Ramanujan's Distinguished Contemporary
\\
Fermat and Ramanujan: A Comparison \\
J. J. Sylvester: Ramanujan's Illustrious Predecessor
\\
Erd{\H{o}}s and Ramanujan: Legends of Twentieth Century
Mathematics \\
C. G. J. Jacobi: Algorist par-excellence \\
{\'E}variste Galois: Founder of Group Theory \\
Leonhard Euler: Most Prolific Mathematician in History
\\
G. H. Hardy: Ramanujan's Mentor \\
J. E. Littlewood: Ramanujan's Contemporary and Hardy's
Collaborator \\
Niels Henrik Abel: Norwegian Mathematical Genius \\
Issai Schur: Ramanujan's German Contemporary \\
Robert Rankin: Scottish Link with Ramanujan \\
Part 2: On Some Aspects of Ramanujan's Mathematics \\
Ramanujan and $\pi$ \\
Ramanujan and Partitions \\
Major Progress on a Problem of Ramanujan \\
Part 3: Book Reviews \\
Genius Whom the Gods Loved --- A Review of ``Srinivasa
Ramanujan: The Lost Notebook and Other Unpublished
Papers'' \\
The Discovery and Rediscovery of Mathematical Genius
--- A Review of ``The Man Who Knew Infinity: A Life of
the Genius Ramanujan'' \\
A Review of ``Ramanujan: Letters and Commentary'' \\
A Review of ``Ramanujan: Essays and Surveys'' \\
A Review of ``Partition: A Play on Ramanujan'' \\
Part 4: Preserving Ramanujan's Legacy \\
The Ramanujan Journal: Its Conception, Need and Place
\\
A Pilgrimage to Ramanujan's Hometown \\
The First SASTRA Ramanujan Prizes \\
Ramanujan's Growing Influence",
}
@Book{Sidoli:2014:ATB,
editor = "Nathan Sidoli and Glen {Van Brummelen}",
booktitle = "From {Alexandria}, Through {Baghdad}: Surveys and
Studies in the {Ancient Greek} and {Medieval Islamic}
Mathematical Sciences in Honor of {J. L. Berggren}",
title = "From {Alexandria}, Through {Baghdad}: Surveys and
Studies in the {Ancient Greek} and {Medieval Islamic}
Mathematical Sciences in Honor of {J. L. Berggren}",
publisher = pub-SV,
address = pub-SV:adr,
pages = "xv + 583",
year = "2014",
DOI = "https://doi.org/10.1007/978-3-642-36736-6",
ISBN = "3-642-36735-6, 3-642-36736-4",
ISBN-13 = "978-3-642-36735-9, 978-3-642-36736-6",
LCCN = "QA21-27",
bibdate = "Tue Mar 4 14:29:47 MST 2014",
bibsource = "fsz3950.oclc.org:210/WorldCat;
https://www.math.utah.edu/pub/tex/bib/pi.bib",
series = "SpringerLink: B{\"u}cher",
URL = "http://scans.hebis.de/HEBCGI/show.pl?33313183_aub.html;
http://scans.hebis.de/HEBCGI/show.pl?33313183_toc.html",
abstract = "This book honors the career of historian of
mathematics J.L. Berggren, his scholarship, and service
to the broader community. The first part, of value to
scholars, graduate students, and interested readers, is
a survey of scholarship in the mathematical sciences in
ancient Greece and medieval Islam. It consists of six
articles (three by Berggren himself) covering research
from the middle of the 20th century to the present. The
remainder of the book contains studies by eminent
scholars of the ancient and medieval mathematical
sciences. They serve both as examples of the breadth of
current approaches and topics, and as tributes to
Berggren's interests by his friends and colleagues.",
acknowledgement = ack-nhfb,
subject = "Mathematics; History; Mathematics, Greek; Mathematics,
Arab; MATHEMATICS / Essays; MATHEMATICS / Pre-Calculus;
MATHEMATICS / Reference",
tableofcontents = "History of Greek Mathematics \\
Mathematical Reconstructions Out, Textual Studies in
\\
Research on Ancient Greek Mathematical Sciences \\
History of Mathematics in the Islamic World \\
Mathematics and Her Sisters in Medieval Islam \\
A Survey of Research in the Mathematical Sciences in
Medieval Islam from 1996 to 2011 \\
The Life of Pi: From Archimedes to ENIAC and Beyond \\
Mechanical Astronomy: A Route to the Ancient Discovery
of Epicycles and Eccentrics \\
Some Greek Sundial Meridians \\
An Archimedean Proof of Heron's Formula for the Area of
a Triangle \\
Reading the Lost Folia of the Archimedean Palimpsest
\\
Acts of geometrical construction in the Spherics of
Theodosios \\
Archimedes Among the Ottomans \\
The `Second' Arabic Translation of Theodosius'
Sphaerica \\
More Archimedean than Archimedes: A New Trace of Abu
Sahl al-Kuhi's work in Latin \\
Les math{\'e}matiques en Occident musulman \\
Ibn al-Raqqam's al-Zij al-Mustawfi in MS Rabat National
Library 2461 \\
An Ottoman astrolabe full of surprises \\
Un alg{\'e}briste arabe: Abu Kamil SuCac ibn Aslam \\
Abu Kamil's Book on Mensuration \\
Hebrew Texts on the Regular Polyhedra \\
A Treatise by Biruni on the Rule of Three and its
Variations \\
Safavid Art, Science, and Courtly Education in the
Seventeenth Century \\
Translating Playfair's Geometry into Arabic.",
}
@Book{Pitici:2017:BWM,
editor = "Mircea Pitici",
booktitle = "The Best Writing on Mathematics",
title = "The Best Writing on Mathematics",
volume = "2016",
publisher = pub-PRINCETON,
address = pub-PRINCETON:adr,
pages = "xxii + 377",
year = "2017",
ISBN = "0-691-17529-2 (paperback)",
ISBN-13 = "978-0-691-17529-4 (paperback)",
LCCN = "QA1 .B337; QA93 .B476 2016",
bibdate = "Tue Nov 20 10:42:49 MST 2018",
bibsource = "fsz3950.oclc.org:210/WorldCat;
https://www.math.utah.edu/pub/tex/bib/einstein.bib;
https://www.math.utah.edu/pub/tex/bib/pi.bib",
abstract = "An anthology of the year's finest writing on
mathematics from around the world, featuring promising
new voices as well as some of the foremost names in
mathematics.",
acknowledgement = ack-nhfb,
subject = "Mathematics",
tableofcontents = "Introduction / Mircea Pitici \\
Mathematics and teaching / Hyman Bass \\
In defense of pure mathematics / Daniel S. Silver \\
G. H. Hardy: mathematical biologist / Hannah Elizabeth
Christenson and Stephan Ramon Garcia \\
The reasonable ineffectiveness of mathematics / Derek
Abbott \\
Stacking wine bottles revisited / Burkard Polster \\
The way the billiard ball bounces / Joshua Bowman \\
The intersection game / Burkhard Polster \\
Tonight! Epic math battles: counting vs. matching /
Jennifer J. Quinn \\
Mathematicians chase moonshine's shadow / Erica
Klarreich \\
The impenetrable proof / Davide Castelvecchi \\
A proof that some spaces can't be cut / Kevin Hartnett
\\
Einstein's first proof / Steven Strogatz \\
Why string theory still offers hope we can unify
physics / Brian Greene \\
The pioneering role of the Sierpinski Gasket / Tanya
Khovanova, Eric Nie, and Alok Puranik \\
Fractals as photographs / Marc Frantz \\
Math at the Met / Joseph Dauben and Marjorie Senechal
\\
Common sense about the common core / Alan H. Schoenfeld
\\
Explaining your math: unnecessary at best, encumbering
at worst / Katharine Beals and Barry Garelick \\
Teaching applied mathematics / David Acheson, Peter R.
Turner, Gilbert Strang, and Rachel Levy \\
Circular reasoning: who first proved that $C$ divided
by $d$ is a constant? / David Richeson \\
A medieval mystery : Nicole Oresme's concept of
curvitas / Isabel M. Serrano and Bogdan D. Suceav\?a
\\
The myth of Leibniz's proof of the fundamental theorem
of calculus / Viktor Bl{\^e}asj{\"o} \\
The spirograph and mathematical models from
Nineteenth-Century Germany / Amy Shell-Gellasch \\
What does ``depth'' mean in mathematics? / John
Stillwell \\
Finding errors in big data / Marco Puts, Piet Daas, and
Ton De Waal \\
Programs and probability / Brian Hayes \\
Lottery perception / Jorge Almeida \\
Why acknowledging uncertainty can make you a better
scientist / Andrew Gelman \\
For want of a nail: why unnecessarily long tests may be
impeding the progress of western civilization / Howard
Wainer and Richard Feinberg \\
How to write a general interest mathematics book / Ian
Stewart",
}
@Proceedings{Bailey:2020:AVC,
editor = "David H. Bailey and Naomi Simone Borwein and Richard
P. Brent and Regina S. Burachik and Judy-anne Heather
Osborn and Brailey Sims and Qiji J. Zhu",
booktitle = "From Analysis to Visualization: A Celebration of the
Life and Legacy of {Jonathan M. Borwein, Callaghan,
Australia, September 2017}",
title = "From Analysis to Visualization: A Celebration of the
Life and Legacy of {Jonathan M. Borwein, Callaghan,
Australia, September 2017}",
volume = "313",
publisher = pub-SV,
address = pub-SV:adr,
year = "2020",
DOI = "https://doi.org/10.1007/978-3-030-36568-4",
ISBN = "3-030-36567-0 (print), 3-030-36568-9 (e-book)",
ISBN-13 = "978-3-030-36567-7 (print), 978-3-030-36568-4
(e-book)",
ISSN = "2194-1009 (print), 2194-1017 (electronic)",
LCCN = "????",
MRclass = "00B20, 11-06, 26-06, 33-06, 47-06, 49-06, 52-06,
62P05, 91G99, 97-06",
bibdate = "Tue Apr 21 10:22:01 MDT 2020",
bibsource = "fsz3950.oclc.org:210/WorldCat;
https://www.math.utah.edu/pub/bibnet/authors/b/borwein-jonathan-m.bib;
https://www.math.utah.edu/pub/tex/bib/agm.bib;
https://www.math.utah.edu/pub/tex/bib/pi.bib",
series = "Springer Proceedings in Mathematics \& Statistics",
ZMnumber = "07174492",
acknowledgement = ack-nhfb,
remark = "Book.",
subject = "Education / Teaching Methods and Materials /
Mathematics; Mathematics / Applied; Mathematics /
Mathematical Analysis; Mathematics / Number Theory",
subject-dates = "Jonathan Michael Borwein (20 May 1951--2 August
2016)",
tableofcontents = "Part I: Applied Analysis, Optimisation, and
Convexity \\
Introduction / Regina S. Burachik and Guoyin Li / 3--5
\\
Symmetry and the Monotonicity of Certain Riemann Sums /
David Borwein and Jonathan M. Borwein and Brailey Sims
/ 7--20 \\
Risk and Utility in the Duality Framework of Convex
Analysis / R. Tyrrell Rockafellar / 21--42 \\
Characterizations of Robust and Stable Duality for
Linearly Perturbed Uncertain Optimization Problems /
Nguyen Dinh and Miguel A. Goberna and Marco A. Lopez
and Michel Volle / 43--74 \\
Comparing Averaged Relaxed Cutters and Projection
Methods: Theory and Examples / Reinier Diaz Millan and
Scott B. Lindstrom and Vera Roshchina / 75--98 \\
Part II: Education \\
Introduction / Naomi Simone Borwein / 101--102 \\
On the Educational Legacies of Jonathan M. Borwein /
Naomi Simone Borwein and Judy-anne Heather Osborn /
103--131 \\
How Mathematicians Learned to Stop Worrying and Love
the Computer / Keith Devlin / 133--139 \\
Crossing Boundaries: Fostering Collaboration Between
Mathematics Educators and Mathematicians in Initial
Teacher Education Programmes / Merrilyn Goos / 141--148
\\
Mathematics Education in the Computational Age:
Challenges and Opportunities / Kathryn Holmes /
149--152 \\
Mathematics Education for Indigenous Students in
Preparation for Engineering and Information
Technologies / Collin Phillips and Fu Ken Ly / 153--169
\\
Origami as a Teaching Tool for Indigenous Mathematics
Education / Michael Assis and Michael Donovan /
171--188 \\
Dynamic Visual Models: Ancient Ideas and New
Technologies / Damir Jungic and Veselin Jungic /
189--201 \\
A Random Walk Through Experimental Mathematics / Eunice
Y. S. Chan and Robert M. Corless / 203--226 \\
Part III: Financial Mathematics \\
Introduction / David H. Bailey and Qiji J. Zhu /
229--231 \\
A Holistic Approach to Empirical Analysis: The
Insignificance of $P$, Hypothesis Testing and
Statistical Significance* / Morris Altman / 233--253
\\
Do Financial Gurus Produce Reliable Forecasts? / David
H. Bailey and Jonathan M. Borwein and Amir Salehipour
and Marcos Lopez de Prado / 255--274 \\
Entropy Maximization in Finance / Jonathan M. Borwein
and Qiji J. Zhu / 275--295 \\
Part IV: Number Theory, Special Functions, and Pi \\
Introduction / Richard P. Brent / 299--302 \\
Binary Constant-Length Substitutions and Mahler
Measures of Borwein Polynomials / Michael Baake and
Michael Coons and Neil Manibo / 303--322 \\
The Borwein Brothers, Pi and the AGM / Richard P. Brent
/ 323--347 \\
The Road to Quantum Computational Supremacy / Cristian
S. Calude and Elena Calude / 349--367 \\
Nonlinear Identities for Bernoulli and Euler
Polynomials / Karl Dilcher / 369--376 \\
Metrical Theory for Small Linear Forms and Applications
to Interference Alignment / Mumtaz Hussain and Seyyed
Hassan Mahboubi and Abolfazl Seyed Motahari / 377--393
\\
Improved Bounds on Brun's Constant / Dave Platt and Tim
Trudgian / 395--406 \\
Extending the PSLQ Algorithm to Algebraic Integer
Relations / Matthew P. Skerritt and Paul Vrbik /
407--421 \\
Short Walk Adventures / Armin Straub and Wadim Zudilin
/ 423--439",
}