%%% -*-BibTeX-*- %%% ==================================================================== %%% BibTeX-file{ %%% author = "Nelson H. F. Beebe", %%% version = "1.92", %%% date = "31 January 2019", %%% time = "15:29:45 MST", %%% 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 = "http://www.math.utah.edu/~beebe", %%% checksum = "04315 9551 44665 432807", %%% 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 1.92, the year coverage looked %%% like this: %%% %%% 1727 ( 1) 1825 ( 0) 1923 ( 0) %%% 1728 ( 0) 1826 ( 0) 1924 ( 1) %%% 1729 ( 0) 1827 ( 0) 1925 ( 1) %%% 1730 ( 0) 1828 ( 0) 1926 ( 2) %%% 1731 ( 0) 1829 ( 0) 1927 ( 1) %%% 1734 ( 0) 1832 ( 0) 1930 ( 1) %%% 1735 ( 0) 1833 ( 0) 1931 ( 1) %%% 1737 ( 0) 1835 ( 0) 1933 ( 1) %%% 1739 ( 0) 1837 ( 0) 1935 ( 1) %%% 1740 ( 0) 1838 ( 0) 1936 ( 1) %%% 1742 ( 0) 1840 ( 0) 1938 ( 2) %%% 1743 ( 0) 1841 ( 0) 1939 ( 2) %%% 1744 ( 0) 1842 ( 0) 1940 ( 1) %%% 1746 ( 0) 1844 ( 0) 1942 ( 1) %%% 1749 ( 0) 1847 ( 0) 1945 ( 1) %%% 1750 ( 0) 1848 ( 0) 1946 ( 2) %%% 1751 ( 0) 1849 ( 0) 1947 ( 2) %%% 1752 ( 0) 1850 ( 0) 1948 ( 1) %%% 1754 ( 0) 1852 ( 0) 1950 ( 6) %%% 1755 ( 0) 1853 ( 1) 1951 ( 0) %%% 1758 ( 0) 1856 ( 0) 1954 ( 1) %%% 1759 ( 0) 1857 ( 0) 1955 ( 4) %%% 1761 ( 0) 1859 ( 0) 1957 ( 1) %%% 1762 ( 0) 1860 ( 0) 1958 ( 1) %%% 1763 ( 0) 1861 ( 0) 1959 ( 1) %%% 1764 ( 0) 1862 ( 0) 1960 ( 2) %%% 1765 ( 0) 1863 ( 0) 1961 ( 1) %%% 1766 ( 0) 1864 ( 0) 1962 ( 4) %%% 1768 ( 1) 1866 ( 0) 1964 ( 0) %%% 1769 ( 0) 1867 ( 0) 1965 ( 1) %%% 1771 ( 0) 1869 ( 0) 1967 ( 3) %%% 1772 ( 0) 1870 ( 0) 1968 ( 1) %%% 1773 ( 0) 1871 ( 2) 1969 ( 3) %%% 1774 ( 0) 1872 ( 0) 1970 ( 2) %%% 1775 ( 0) 1873 ( 1) 1971 ( 2) %%% 1776 ( 0) 1874 ( 0) 1972 ( 1) %%% 1777 ( 0) 1875 ( 0) 1973 ( 1) %%% 1780 ( 0) 1878 ( 0) 1976 ( 5) %%% 1781 ( 0) 1879 ( 1) 1977 ( 1) %%% 1782 ( 0) 1880 ( 0) 1978 ( 3) %%% 1783 ( 0) 1881 ( 0) 1979 ( 3) %%% 1784 ( 0) 1882 ( 1) 1980 ( 2) %%% 1785 ( 0) 1883 ( 1) 1981 ( 3) %%% 1786 ( 0) 1884 ( 0) 1982 ( 2) %%% 1787 ( 0) 1885 ( 0) 1983 ( 3) %%% 1788 ( 0) 1886 ( 0) 1984 ( 4) %%% 1789 ( 0) 1887 ( 0) 1985 ( 3) %%% 1790 ( 0) 1888 ( 0) 1986 ( 7) %%% 1791 ( 0) 1889 ( 0) 1987 ( 4) %%% 1792 ( 0) 1890 ( 0) 1988 ( 8) %%% 1793 ( 0) 1891 ( 1) 1989 ( 6) %%% 1794 ( 0) 1892 ( 0) 1990 ( 4) %%% 1795 ( 0) 1893 ( 0) 1991 ( 4) %%% 1796 ( 0) 1894 ( 0) 1992 ( 4) %%% 1797 ( 0) 1895 ( 1) 1993 ( 4) %%% 1798 ( 0) 1896 ( 1) 1994 ( 6) %%% 1799 ( 0) 1897 ( 0) 1995 ( 4) %%% 1800 ( 0) 1898 ( 0) 1996 ( 6) %%% 1801 ( 0) 1899 ( 0) 1997 ( 13) %%% 1802 ( 0) 1900 ( 0) 1998 ( 6) %%% 1803 ( 0) 1901 ( 0) 1999 ( 5) %%% 1804 ( 0) 1902 ( 0) 2000 ( 9) %%% 1805 ( 0) 1903 ( 0) 2001 ( 6) %%% 1806 ( 0) 1904 ( 1) 2002 ( 4) %%% 1807 ( 0) 1905 ( 0) 2003 ( 5) %%% 1808 ( 0) 1906 ( 0) 2004 ( 9) %%% 1809 ( 0) 1907 ( 0) 2005 ( 5) %%% 1810 ( 0) 1908 ( 0) 2006 ( 4) %%% 1811 ( 0) 1909 ( 0) 2007 ( 1) %%% 1812 ( 0) 1910 ( 0) 2008 ( 9) %%% 1813 ( 0) 1911 ( 0) 2009 ( 3) %%% 1814 ( 0) 1912 ( 0) 2010 ( 12) %%% 1815 ( 0) 1913 ( 0) 2011 ( 16) %%% 1816 ( 0) 1914 ( 1) 2012 ( 6) %%% 1817 ( 0) 1915 ( 0) 2013 ( 14) %%% 1818 ( 0) 1916 ( 0) 2014 ( 6) %%% 1819 ( 0) 1917 ( 0) 2015 ( 7) %%% 1820 ( 0) 1918 ( 0) 2016 ( 2) %%% 1821 ( 0) 1919 ( 0) 2017 ( 4) %%% 1822 ( 0) 1920 ( 0) 2018 ( 3) %%% 1823 ( 0) 1921 ( 1) %%% %%% Article: 211 %%% Book: 32 %%% InBook: 1 %%% InCollection: 8 %%% InProceedings: 10 %%% Misc: 11 %%% Proceedings: 4 %%% TechReport: 11 %%% Unpublished: 19 %%% %%% Total entries: 307 %%% %%% 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 entry Yee:2013:IST %%% for the latest record. 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|http://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-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-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-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-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-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-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 = "http://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 = "http://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; http://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 = "http://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 = "http://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 = "http://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 = "http://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 = "http://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 $$z^{n} + a_1z^{n-1}\ldots + a_n = 0$$ 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 $$0 = N_0e^{z_0} + N_1e^{z_1} + \cdots + N_n e^{z_n}$$ 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 = "http://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 = "http://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 = "http://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 = "http://www.jstor.org/page/journal/amermathmont/about.html", } @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 = "http://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 = "http://www.jstor.org/page/journal/amermathmont/about.html", } @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 = "http://www.math.utah.edu/pub/tex/bib/pi.bib; JSTOR database", acknowledgement = ack-nhfb, fjournal = "American Mathematical Monthly", journal-URL = "http://www.jstor.org/page/journal/amermathmont/about.html", } @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 = "http://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 = "http://www.math.utah.edu/pub/tex/bib/pi.bib; JSTOR database", acknowledgement = ack-nhfb, fjournal = "American Mathematical Monthly", journal-URL = "http://www.jstor.org/page/journal/amermathmont/about.html", } @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 = "http://www.math.utah.edu/pub/tex/bib/pi.bib; JSTOR database", acknowledgement = ack-nhfb, fjournal = "American Mathematical Monthly", journal-URL = "http://www.jstor.org/page/journal/amermathmont/about.html", } @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 = "http://www.math.utah.edu/pub/tex/bib/pi.bib; JSTOR database", acknowledgement = ack-nhfb, fjournal = "American Mathematical Monthly", journal-URL = "http://www.jstor.org/page/journal/amermathmont/about.html", } @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 = "http://www.math.utah.edu/pub/tex/bib/pi.bib; JSTOR database", acknowledgement = ack-nhfb, fjournal = "American Mathematical Monthly", journal-URL = "http://www.jstor.org/page/journal/amermathmont/about.html", } @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 = "http://www.math.utah.edu/pub/tex/bib/pi.bib; JSTOR database", acknowledgement = ack-nhfb, fjournal = "American Mathematical Monthly", journal-URL = "http://www.jstor.org/page/journal/amermathmont/about.html", } @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 = "http://www.math.utah.edu/pub/tex/bib/pi.bib; z3950.loc.gov:7090/Voyager", acknowledgement = ack-nhfb, author-dates = "1887--1920", subject = "Mathematics", remark = "See entry \cite{Hardy:2000:CPS} for table of contents of reprinted edition.", } @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 = "http://www.math.utah.edu/pub/tex/bib/pi.bib; JSTOR database", acknowledgement = ack-nhfb, fjournal = "American Mathematical Monthly", journal-URL = "http://www.jstor.org/page/journal/amermathmont/about.html", } @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 = "http://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 = "http://www.math.utah.edu/pub/tex/bib/pi.bib; JSTOR database", acknowledgement = ack-nhfb, fjournal = "American Mathematical Monthly", journal-URL = "http://www.jstor.org/page/journal/amermathmont/about.html", } @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 = "http://www.math.utah.edu/pub/tex/bib/pi.bib; JSTOR database", acknowledgement = ack-nhfb, fjournal = "American Mathematical Monthly", journal-URL = "http://www.jstor.org/page/journal/amermathmont/about.html", } @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; http://www.math.utah.edu/pub/tex/bib/osiris.bib; http://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 = "http://www.math.utah.edu/pub/tex/bib/pi.bib; JSTOR database", acknowledgement = ack-nhfb, fjournal = "American Mathematical Monthly", journal-URL = "http://www.jstor.org/page/journal/amermathmont/about.html", } @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 = "http://www.math.utah.edu/pub/tex/bib/pi.bib; JSTOR database", acknowledgement = ack-nhfb, fjournal = "American Mathematical Monthly", journal-URL = "http://www.jstor.org/page/journal/amermathmont/about.html", } @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 = "http://www.math.utah.edu/pub/tex/bib/pi.bib; JSTOR database", acknowledgement = ack-nhfb, fjournal = "American Mathematical Monthly", journal-URL = "http://www.jstor.org/page/journal/amermathmont/about.html", } @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 = "http://www.math.utah.edu/pub/tex/bib/pi.bib; JSTOR database", acknowledgement = ack-nhfb, fjournal = "American Mathematical Monthly", journal-URL = "http://www.jstor.org/page/journal/amermathmont/about.html", } @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 = "http://www.math.utah.edu/pub/tex/bib/pi.bib; JSTOR database", acknowledgement = ack-nhfb, fjournal = "American Mathematical Monthly", journal-URL = "http://www.jstor.org/page/journal/amermathmont/about.html", } @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 = "http://www.math.utah.edu/pub/tex/bib/pi.bib; JSTOR database", acknowledgement = ack-nhfb, fjournal = "American Mathematical Monthly", journal-URL = "http://www.jstor.org/page/journal/amermathmont/about.html", } @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 = "http://www.math.utah.edu/pub/tex/bib/pi.bib; JSTOR database", acknowledgement = ack-nhfb, fjournal = "American Mathematical Monthly", journal-URL = "http://www.jstor.org/page/journal/amermathmont/about.html", } @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 = "http://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 = "http://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 = "http://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 = "http://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 $pi/4$ 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 = "http://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 = "http://www.math.utah.edu/pub/tex/bib/pi.bib; http://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 = "http://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 = "http://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 = "http://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 = "http://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 = "http://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 = "http://www.math.utah.edu/pub/tex/bib/pi.bib; JSTOR database", acknowledgement = ack-nhfb, fjournal = "American Mathematical Monthly", journal-URL = "http://www.jstor.org/page/journal/amermathmont/about.html", } @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 = "http://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 = "http://www.math.utah.edu/pub/tex/bib/pi.bib; JSTOR database", acknowledgement = ack-nhfb, fjournal = "American Mathematical Monthly", journal-URL = "http://www.jstor.org/page/journal/amermathmont/about.html", } @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 = "http://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 = "http://www.math.utah.edu/pub/tex/bib/pi.bib; JSTOR database", acknowledgement = ack-nhfb, fjournal = "American Mathematical Monthly", journal-URL = "http://www.jstor.org/page/journal/amermathmont/about.html", } @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", pages = "12--17", year = "1957", LCCN = "QA11.A1 O9 1957", bibdate = "Fri Jul 1 09:32:16 MDT 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/pi.bib; library.ox.ac.uk:210/ADVANCE", note = "Footnote 12-53.", acknowledgement = ack-nhfb, bookpages = "111", 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", year = "1958", LCCN = "????", bibdate = "Sat Jan 07 16:58:57 2012", bibsource = "http://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, bookpages = "????", 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 = "http://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 = "http://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 = "http://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 = "http://www.math.utah.edu/pub/tex/bib/pi.bib; JSTOR database", acknowledgement = ack-nhfb, fjournal = "American Mathematical Monthly", journal-URL = "http://www.jstor.org/page/journal/amermathmont/about.html", } @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 = "http://www.math.utah.edu/pub/tex/bib/pi.bib; JSTOR database", acknowledgement = ack-nhfb, fjournal = "American Mathematical Monthly", journal-URL = "http://www.jstor.org/page/journal/amermathmont/about.html", } @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 = "http://www.math.utah.edu/pub/tex/bib/pi.bib; z3950.loc.gov:7090/Voyager", acknowledgement = ack-nhfb, author-dates = "1887--1920", subject = "Mathematics", remark = "See entry \cite{Hardy:2000:CPS} for table of contents of reprinted edition.", } @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 = "http://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 = "http://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{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 = "http://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 = "http://www.math.utah.edu/pub/tex/bib/pi.bib; http://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; http://www.math.utah.edu/pub/tex/bib/compj2010.bib; http://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/; http://www.math.utah.edu/pub/tex/bib/cacm1960.bib; http://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 = "http://www.math.utah.edu/pub/tex/bib/pi.bib; http://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 = "http://www.math.utah.edu/pub/tex/bib/pi.bib; JSTOR database", acknowledgement = ack-nhfb, fjournal = "American Mathematical Monthly", journal-URL = "http://www.jstor.org/page/journal/amermathmont/about.html", } @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; http://www.math.utah.edu/pub/tex/bib/fibquart.bib; http://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 = "http://www.math.utah.edu/pub/tex/bib/pi.bib; JSTOR database", acknowledgement = ack-nhfb, fjournal = "American Mathematical Monthly", journal-URL = "http://www.jstor.org/page/journal/amermathmont/about.html", } @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 = "http://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; http://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 = "http://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 = "http://www.math.utah.edu/pub/tex/bib/mathcomp1970.bib; http://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 = "http://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 = "http://www.math.utah.edu/pub/tex/bib/pi.bib; JSTOR database", acknowledgement = ack-nhfb, fjournal = "American Mathematical Monthly", journal-URL = "http://www.jstor.org/page/journal/amermathmont/about.html", } @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; http://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 $O(M(n)log (n))$ 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 = "http://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 = "http://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 = "http://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 = "http://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; http://www.math.utah.edu/pub/tex/bib/fibquart.bib; http://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 = "http://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 = "http://www.math.utah.edu/pub/tex/bib/pi.bib; http://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 = "http://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 = "http://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 = "http://www.math.utah.edu/pub/tex/bib/pi.bib; JSTOR database", acknowledgement = ack-nhfb, fjournal = "American Mathematical Monthly", journal-URL = "http://www.jstor.org/page/journal/amermathmont/about.html", } @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 = "http://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 = "http://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 = "http://www.math.utah.edu/pub/tex/bib/pi.bib; JSTOR database", acknowledgement = ack-nhfb, fjournal = "American Mathematical Monthly", journal-URL = "http://www.jstor.org/page/journal/amermathmont/about.html", } @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 = "http://www.math.utah.edu/pub/tex/bib/histmath.bib; http://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 = "http://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 = "http://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", MRnumber = "MR0654686 (83e:41021)", MRclass = "41A21", address = "Montr{\'e}al, QC H3C 3J7, Canada", pages = "149--174", year = "1982", MRnumber = "41A20 41A25 65D20", series = "S{\'e}minaire de Math{\'e}matiques Sup{\'e}rieures", ZMnumber = "0506.41014", bibdate = "Thu Jun 09 15:53:59 2016", bibsource = "http://www.math.utah.edu/pub/tex/bib/pi.bib", 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; http://www.math.utah.edu/pub/tex/bib/bit.bib; http://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, fjournal = "BIT", journal-URL = "http://link.springer.com/journal/10543", } @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 = "http://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 = "http://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/; http://www.math.utah.edu/pub/tex/bib/pi.bib", acknowledgement = ack-nhfb, fjournal = "Canadian mathematical bulletin = Bulletin canadien de math{\'e}matiques", journal-URL = "http://cms.math.ca/cmb/", } @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 = "http://www.math.utah.edu/pub/bibnet/authors/b/borwein-jonathan-m.bib; http://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, } @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 = "http://www.math.utah.edu/pub/tex/bib/mathcomp1980.bib; http://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 = "http://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 = "http://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 = "JSTOR database; http://www.math.utah.edu/pub/tex/bib/mathcomp1980.bib; http://www.math.utah.edu/pub/tex/bib/pi.bib", 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; http://www.math.utah.edu/pub/tex/bib/bit.bib; http://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, fjournal = "BIT", journal-URL = "http://link.springer.com/journal/10543", } @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 = "http://www.math.utah.edu/pub/tex/bib/pi.bib; JSTOR database", acknowledgement = ack-nhfb, 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", 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 = "http://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 = "http://www.math.utah.edu/pub/tex/bib/pi.bib; JSTOR database", acknowledgement = ack-nhfb, fjournal = "American Mathematical Monthly", journal-URL = "http://www.jstor.org/page/journal/amermathmont/about.html", } @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 = "http://www.math.utah.edu/pub/tex/bib/pi.bib; JSTOR database", acknowledgement = ack-nhfb, fjournal = "American Mathematical Monthly", journal-URL = "http://www.jstor.org/page/journal/amermathmont/about.html", } @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 = "http://www.math.utah.edu/pub/tex/bib/pi.bib; JSTOR database", acknowledgement = ack-nhfb, fjournal = "American Mathematical Monthly", journal-URL = "http://www.jstor.org/page/journal/amermathmont/about.html", } @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 = "http://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 = "http://www.math.utah.edu/pub/tex/bib/pi.bib; JSTOR database", acknowledgement = ack-nhfb, fjournal = "American Mathematical Monthly", journal-URL = "http://www.jstor.org/page/journal/amermathmont/about.html", } @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 = "http://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 = "http://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 = "http://www.math.utah.edu/pub/tex/bib/pi.bib; JSTOR database", acknowledgement = ack-nhfb, fjournal = "American Mathematical Monthly", journal-URL = "http://www.jstor.org/page/journal/amermathmont/about.html", } @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/10.2307/2007932", 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 = "http://www.math.utah.edu/pub/tex/bib/pi.bib; 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", 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 = "http://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", 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 = "http://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 = "http://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; http://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 = "http://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 (paper), 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 = "http://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", 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); http://www.math.utah.edu/pub/tex/bib/pi.bib", acknowledgement = ack-nhfb, affiliation = "Dalhousie Univ., Halifax; Dalhousie Univ., Halifax", 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 = "http://www.jstor.org/page/journal/amermathmont/about.html", journalabbrev = "Am. Math. Monthly", jrldate = "March 1989", 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 = "http://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; http://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 = "http://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; http://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 ($\approx 3.1416408R$).", 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; http://www.math.utah.edu/pub/tex/bib/jsuper.bib; http://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", 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 = "http://www.math.utah.edu/pub/tex/bib/pi.bib; JSTOR database", acknowledgement = ack-nhfb, fjournal = "American Mathematical Monthly", journal-URL = "http://www.jstor.org/page/journal/amermathmont/about.html", } @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; http://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 = "http://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", } @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 = "http://www.math.utah.edu/pub/tex/bib/pi.bib; JSTOR database", acknowledgement = ack-nhfb, fjournal = "American Mathematical Monthly", journal-URL = "http://www.jstor.org/page/journal/amermathmont/about.html", } @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 = "http://www.math.utah.edu/pub/tex/bib/jcomputapplmath1990.bib; http://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 = "http://www.math.utah.edu/pub/bibnet/authors/h/heisenberg-werner.bib; http://www.math.utah.edu/pub/bibnet/authors/m/mandelbrot-benoit.bib; http://www.math.utah.edu/pub/bibnet/authors/s/shannon-claude-elwood.bib; http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/fibquart.bib; http://www.math.utah.edu/pub/tex/bib/master.bib; http://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; http://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 = "http://www.math.utah.edu/pub/tex/bib/master.bib; http://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 = "http://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 $: The {Gauss--Salamin} algorithm", journal = j-MATH-GAZ, volume = "76", number = "476", pages = "231--242", year = "1992", CODEN = "MAGAAS", DOI = "https://doi.org/10.2307/3619132", ISSN = "0025-5572; 2056-6328/e", MRnumber = "11Y16 11-01 33E05", bibdate = "Thu Jun 9 15:49:28 2016", bibsource = "http://www.math.utah.edu/pub/tex/bib/pi.bib", ZMnumber = "0805.11086", acknowledgement = ack-nhfb, fjournal = "The Mathematical Gazette", journal-URL = "http://www.m-a.org.uk/jsp/index.jsp?lnk=620", language = "English", } @Book{Mauron:1992:P, author = "C. Mauron", title = "$\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 = "http://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 = "http://www.math.utah.edu/pub/tex/bib/histmath.bib; http://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 = "http://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; http://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", 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 = "http://www.math.utah.edu/pub/tex/bib/master.bib; http://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 = "http://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, remark = "The author writes: ``I will \ldots{} virtually rule out any possibility that Lazzarini performed a valid experiment [of Buffon needle-casting].''", keywords = "Buffon needle approximation of $\pi$", fjournal = "Mathematics Magazine", journal-URL = "http://www.maa.org/pubs/mathmag.html", } @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 = "http://www.math.utah.edu/pub/tex/bib/pi.bib", URL = "http://www.ams.org/mathscinet-getitem?mr=96e:11168", acknowledgement = ack-nhfb, fjournal = "Experimental Mathematics", journal-URL = "http://www.tandfonline.com/loi/uexm20", } @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 = "http://www.math.utah.edu/pub/tex/bib/master.bib; http://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; http://www.math.utah.edu/pub/tex/bib/fibquart.bib; http://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 = "http://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 = "http://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 = "http://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", 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 = "http://www.math.utah.edu/pub/bibnet/authors/b/borwein-jonathan-m.bib; http://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, 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 = "http://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 = "http://www.math.utah.edu/pub/tex/bib/pi.bib", acknowledgement = ack-nhfb, fjournal = "American Mathematical Monthly", journal-URL = "http://www.jstor.org/page/journal/amermathmont/about.html", } @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 = "http://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 = "http://www.math.utah.edu/pub/tex/bib/master.bib; http://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 = "http://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 = "http://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 = "http://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 = "http://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 = "http://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 = "http://www.jstor.org/page/journal/amermathmont/about.html", } @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 = "http://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 = "http://www.jstor.org/page/journal/amermathmont/about.html", } @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 = "http://www.math.utah.edu/pub/tex/bib/pi.bib", acknowledgement = ack-nhfb, fjournal = "Mathematical Intelligencer", } @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; http://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 = "$\pi$", } @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 = "http://www.math.utah.edu/pub/tex/bib/pi.bib", series = "CMS Conference Proceedings", acknowledgement = ack-nhfb, } @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 = "http://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", } @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 = "http://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 = "http://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{Berggren:1997:PSB, editor = "J. L. 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", ISBN = "0-387-94924-0", ISBN-13 = "978-0-387-94924-6", LCCN = "QA484 .P5 1997", bibdate = "Tue Aug 16 08:54:25 MDT 2016", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/b/borwein-jonathan-m.bib; http://www.math.utah.edu/pub/tex/bib/pi.bib; z3950.loc.gov:7090/Voyager", abstract = "This book documents the 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 most whimsical aspects of mathematics. Much significant mathematics originates with pi, and many great mathematicians have contributed to this story's unfolding. Mathematicians and historians of mathematics will find this book indispensable. Teachers at every level can find here ample resources for anything from individual talks and student projects to special topics courses.", acknowledgement = ack-nhfb, subject = "Pi", tableofcontents = "1. The Rhind Mathematical Papyrus-Problem 50 (ca. 1650 B.C.) \\ 2. Engels, Quadrature of the Circles in Ancient Egypt (1977) \\ 3. Archimedes, Measurement of a Circle (ca. 250 BC) \\ 4. Phillips. Archimedes the Numerical Analysis (1981) \\ 5. Lam and Ang. Circle Measurements in Ancient China (1986) \\ 6. The Banu Musa: The Measurement of Plane and Solid Figures (ca. 850) \\ 7. Madhava. The Power Series for Arctan and Pi (ca. 1400) \\ 8. Hope-Jones. Ludolph (or Ludolff or Lucius) van Ceulen (1938) \\ 9. Viete. Variorum de Rebus Mathematicis Reponsorum Liber VII (1593) \\ 10. Wallis. Computation of $\pi$ by Successive Interpolations (1655) \\ 11. Wallis. Arithmetica Infinitorum (1655) \\ 12. Huygens. De Circull Magnitudine Inventa (1724) \\ 13. Gregory. Correspondence with John Collins (1671) \\ 14. Roy. The Discovery of the Series Formula for $\pi$ by Leibniz, Gregory, and Nilakantha (1990) \\ 15. Jones. The First Use of $\pi$ for the Circle Ratio (1706) \\ 16. Newton. Of the Method of Fluxious and Infinite Series (1737) \\ 17. Euler. Chapter 10 of Introduction to Analysis of the Infinite (On the Use of Discovered Fractions to Sum Infinite Series) (1748) \\ 18. Lambert. M{\'e}moire sur Quelques Propri{\'e}t{\'e}s Remarquables des Quantit{\'e}s Transcendentes Circulaires et Logarithmiques (1761) \\ 19. Lambert. Irrationality of $\pi$ (1969) \\ 20. Shanks. Contributions to Mathematics Comprising Chiefly of the Rectification of the Circle to 607 Places of Decimals (1853) \\ 21. Hermite. Sur La Fonction Exponentielle (1873) \\ 22. Lindemann. Ueber die Zahl $\pi$ (1882) \\ 23. Weierstrass. Zu Lindemann's Abhandlung ``{\"U}ber die Ludolphsche Zahl'' (1885) \\ 24. Hilbert. Ueber die Trancendenz der Zahlen $e$ und $\pi$ (1893) \\ 25. Goodwin. Quadrature of the Circle (1894) \\ 26. Edington. House Bill No. 246, Indiana State Legislature, 1897 (1935) \\ 27. Singmaster. The Legal Values of Pi (1985) \\ 28. Ramanujan. Squaring the Circle (1913) \\ 29. Ramanujan. Modular Equations and Approximations to $\pi$ (1914) \\ 30. Watson. The Marquis and the Land Agent: A Tale of the Eighteenth Century (1933) \\ 31. Ballantine. The Best (?) Formula for Computing $\pi$ to a Thousand Places (1939) \\ 32. Birch. An Algorithm for Construction of Arctangent Relations (1946) \\ 33. Niven. A Simple Proof that $\pi$ Is Irrational (1947) \\ 34. Reitwiesner. An ENIAC Determination of $\pi$ and e to 2000 Decimal Places (1950) \\ 35. Schepler. The Chronology of Pi (1950) \\ 36. Mahler. On the Approximation of $\pi$ (1953) \\ 37. Wrench, Jr. The Evolution of Extended Decimal Approximations to $\pi$ (1960) \\ 38. Shanks and Wrench, Jr. Calculation of $\pi$ to 100,000 Decimals (1962) \\ 39. Sweeny. On the Computation of Euler's Constant (1963) \\ 40. Baker. Approximation to the Logarithms of Certain Rational Numbers (1964) \\ 41. Adams. Asymptotic Diophantine Approximations to E (1966) \\ 42. Mahler. Applications of Some Formulae by Hermite to the Approximations of Exponentials of Logarithms (1967) \\ 43. Eves. In Mathematical Circles; A Selection of Mathematical Stories and Anecdotes (excerpt) (1969) \\ 44. Eves. Mathematical Circles Revisited; A Second Collection of Mathematical Stories and Anecdotes (excerpt) (1971) \\ 45. Todd. The Lemniscate Constants (1975) \\ 46. Salamin. Computation of $\pi$ using Arithmetic--Geometric Mean (1976) \\ 47. Brent. Fast Multiple-Precision Evaluation of Elementary Functions (1976) \\ 48. Beukers. A Note on the Irrationality of $\zeta(2)$ and $\zeta(3)$ (1979) \\ 49. van der Poorten. A Proof that Euler Missed \ldots{} Apery's Proof of the Irrationality of $\zeta(3)$ (1979) \\ 50. Brent and McMillan. Some New Algorithms for High-Precision Computation of Euler's Constant (1980) \\ 51. Apostol. A Proof that Euler Missed: Evaluating $\zeta(2)$ the Easy Way (1983) 52. O'Shaughnessy. Putting God Back in Math (1983) \\ 53. Stern. A Remarkable Approximation to $\pi$ (1985) \\ 54. Newman and Shanks. On a Sequence Arising in Series for $\pi$ (1984) \\ 55. Cox. The Arithmetic--Geometric Mean of Gauss (1984) \\ 56. Borwein and Borwein. The Arithmetic--Geometric Mean and Fast Computation of Elementary Functions (1984) \\ 57. Newmann. A Simplified Version of the Fast Algorithms of Brent and Salamin (1984) \\ 58. Wagon. Is Pi Normal? (1985) \\ 59. Keith. Circle Digits: A Self-Referential Story (1986) \\ 60. Bailey. The Computation of $\pi$ to 29,360,000 Decimal Digits Using Borweins' Quartically Convergent Algorithm (1988) \\ 61. Kanada. Vectorization of Multiple-Precision Arithmetic Program and 201,326,000 Decimal Digits of $\pi$ Calculation (1988) \\ 62. Borwein and Borwein. Ramanujan and Pi (1988) \\ 63. Chudnovsky and Chudnovsky. Approximation and Complex Multiplication According to Ramanujan (1988) \\ 64. Borwein, Borwein and Bailey. Ramanujan, Modular Equations, and Approximations to Pi or How to Compute One Billion Digits of Pi (1989) \\ 65. Borwein, Borwein and Dilcher. Pi, Euler Numbers, and Asymptotic Expansions (1989) \\ 66. Beukers, Bezivin, and Robba. An Alternative Proof of the Lindemann-Weierstrass Theorem (1990) \\ 67. Webster. The Tail of Pi (1991) \\ 68. Eco. An excerpt from Foucault's Pendulum (1993) \\ 69. Keith. Pi Mnemonics and the Art of Constrained Writing (1996) \\ 70. Bailey, Borwein, and Plouffe. On the Rapid Computation of Various Polylogarithmic Constants (1996) \\ App. I. On the Early History of Pi \\ App. II. A Computational Chronology of Pi \\ App. III. Selected Formulae for Pi", } @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 = "http://www.math.utah.edu/pub/tex/bib/master.bib; http://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", } @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 = "http://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 = "http://www.jstor.org/page/journal/amermathmont/about.html", } @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 = "http://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 = "http://www.math.utah.edu/pub/tex/bib/histmath.bib; http://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", } @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/; http://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", } @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 = "http://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, } @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/; http://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", 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 = "http://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", bookpages = "xvii + 273", 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 = "http://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 = "http://www.math.utah.edu/pub/tex/bib/histmath.bib; http://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; http://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/", } @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 = "http://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 = "http://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 = "http://www.math.utah.edu/pub/tex/bib/pi.bib", acknowledgement = ack-nhfb, fjournal = "American Mathematical Monthly", journal-URL = "http://www.jstor.org/page/journal/amermathmont/about.html", } @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)", fjournal = "Mathematical Gazette", journal-URL = "http://www.m-a.org.uk/jsp/index.jsp?lnk=620", bibdate = "Thu Jun 9 15:45:09 2016", acknowledgement = ack-nhfb, bibsource = "http://www.math.utah.edu/pub/tex/bib/pi.bib", } @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 = "http://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", } @Book{Berggren:2000:PSB, author = "Lennart Berggren and Jonathan Borwein and Peter Borwein", 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 = "http://www.math.utah.edu/pub/bibnet/authors/b/borwein-jonathan-m.bib; http://www.math.utah.edu/pub/tex/bib/agm.bib; http://www.math.utah.edu/pub/tex/bib/master.bib; http://www.math.utah.edu/pub/tex/bib/mathcw.bib; http://www.math.utah.edu/pub/tex/bib/pi.bib", acknowledgement = ack-nhfb, libnote = "Not yet in my library.", ORCID-numbers = "Borwein, Jonathan/0000-0002-1263-0646", subject = "Pi (mathematical constant)", tableofcontents = "Preface \\ Acknowledgements \\ Introduction \\ The Rhind Mathematical Papyrus-Problem 50 \\ Engles / Quadrature of the Circle in Ancient Egypt \\ Archimedes / Measurement of a Circle \\ Phillips / Archimedes the Numerical Analyst \\ Lam \& Ang / Circle Measurements in Ancient China \\ The Banu Musa: The Measurement of Plane and Solid Figures \\ Madhava's / The Power Series for Arctan and Pi \\ Hope-Jones / Ludolph van Ceulen \\ Vi{\`e}te / Variorum de Revus Mathematicis Reponsorum Liber VII \\ Wallis / Computation of Pi by Successive Interpolations \\ Wallis / Arithmetica Infinitorum \\ Huygens / De Circuli Magnitudine Inventa \\ Gregory / Correspondence with John Collins \\ Jones / The First Use of Pi for the Circle Ratio \\ Newton / Of The Method of Fluxions and Infinite Series \\ Euler / Chapter 10 of Introduction to Analysis of the Infinite \\ Lambert / M{\'e}moire Sur Quelques Propri{\'e}t{\'e}s Remarquables Des Quantit{\'e}s Transcendentes Circulaires et Logarithmiques \\ Lambert / Irrationality of Pi \\ Shanks / Contributions to Mathematics Comprising Chiefly of the Rectification of the Circle to 607 Places of Decimals \\ Hermite / Sur La Fonction Exponentielle", } @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; http://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; http://www.math.utah.edu/pub/tex/bib/amstat2000.bib; http://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; http://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 = "http://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 = "http://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 = "http://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 = "http://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 = "http://www.math.utah.edu/pub/bibnet/authors/b/borwein-jonathan-m.bib; http://www.math.utah.edu/pub/tex/bib/master.bib; http://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; http://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/", } @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 = "http://www.math.utah.edu/pub/tex/bib/pi.bib", acknowledgement = ack-nhfb, fjournal = "Experimental mathematics", journal-URL = "http://www.tandfonline.com/loi/uexm20", } @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 = "http://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 = "http://www.math.utah.edu/pub/bibnet/authors/c/crandall-richard-e.bib; http://www.math.utah.edu/pub/tex/bib/pi.bib; http://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 = "http://www.math.utah.edu/pub/bibnet/authors/c/crandall-richard-e.bib; http://www.math.utah.edu/pub/tex/bib/pi.bib", note = "pnl.gov", URL = "http://www.pnl.gov/energyscience/10-01/art3.htm; http://web.archive.org/web/20050208141708/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 = "http://www.math.utah.edu/pub/tex/bib/pi.bib", acknowledgement = ack-nhfb, fjournal = "Experimental mathematics", journal-URL = "http://www.tandfonline.com/loi/uexm20", } @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 = "http://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 = "http://www.math.utah.edu/pub/tex/bib/jucs.bib; http://www.math.utah.edu/pub/tex/bib/pi.bib; http://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 = "http://www.math.utah.edu/pub/tex/bib/fparith.bib; http://www.math.utah.edu/pub/tex/bib/ieeeannhistcomput.bib; http://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; http://www.math.utah.edu/pub/tex/bib/expmath.bib; http://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 $m = 8k$, $ 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 = "http://www.math.utah.edu/pub/tex/bib/pi.bib", price = "US\$49.00", acknowledgement = ack-nhfb, } @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 = "http://www.math.utah.edu/pub/tex/bib/master.bib; http://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 = "http://www.math.utah.edu/pub/bibnet/authors/c/crandall-richard-e.bib; http://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 = "http://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 = "http://www.math.utah.edu/pub/tex/bib/pi.bib", acknowledgement = ack-nhfb, fjournal = "Journal of Number Theory {Bordeaux}", } @Book{Berggren:2004:PSB, editor = "Lennart Berggren and Jonathan Borwein and Peter Borwein", title = "Pi, a source book", publisher = pub-SV, address = pub-SV:adr, edition = "Third", pages = "xix + 797", year = "2004", ISBN = "0-387-20571-3", ISBN-13 = "978-0-387-20571-7", LCCN = "QA484 .P5 2004", bibdate = "Sat Apr 23 09:59:19 MDT 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/pi.bib; z3950.loc.gov:7090/Voyager", URL = "http://www.loc.gov/catdir/enhancements/fy0818/2003066023-d.html; http://www.loc.gov/catdir/enhancements/fy0818/2003066023-t.html", acknowledgement = ack-nhfb, remark = "Fourth edition expected in 2011.", subject = "Pi", } @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 = "http://www.math.utah.edu/pub/tex/bib/pi.bib", acknowledgement = ack-nhfb, fjournal = "Canadian Journal of Mathematics = Journal canadien de math{\'e}matiques", journal-URL = "http://cms.math.ca/cjm/", 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 = "http://www.math.utah.edu/pub/tex/bib/pi.bib", price = "US\$45.00", acknowledgement = ack-nhfb, 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 = "http://www.math.utah.edu/pub/tex/bib/computmathappl2000.bib; http://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 = "http://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 = "http://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 = "http://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", } @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; http://www.math.utah.edu/pub/tex/bib/expmath.bib; http://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 = "http://www.math.utah.edu/pub/tex/bib/jstatcomputsimul.bib; http://www.math.utah.edu/pub/tex/bib/pi.bib; http://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 = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib; http://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 = "http://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 = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib; http://www.math.utah.edu/pub/tex/bib/pi.bib; http://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", } @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; http://www.math.utah.edu/pub/tex/bib/fibquart.bib; http://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; http://www.math.utah.edu/pub/tex/bib/expmath.bib; http://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; http://www.math.utah.edu/pub/tex/bib/expmath.bib; http://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 = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib; http://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 = "http://www.math.utah.edu/pub/tex/bib/pi.bib; z3950.loc.gov:7090/Voyager", acknowledgement = ack-nhfb, 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 = "http://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 = "http://www.math.utah.edu/pub/tex/bib/pi.bib", URL = "http://www.carma.newcastle.edu.au/~jb616/pi-2010.pdf", acknowledgement = ack-nhfb, language = "Italian", } @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; http://www.math.utah.edu/pub/tex/bib/fibquart.bib; http://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 = "http://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; http://www.math.utah.edu/pub/tex/bib/amermathmonthly2000.bib; http://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 = "http://www.jstor.org/page/journal/amermathmont/about.html", } @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 = "http://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; http://www.math.utah.edu/pub/tex/bib/amermathmonthly2000.bib; http://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 = "http://www.jstor.org/page/journal/amermathmont/about.html", } @Article{Pickett:2008:ACF, author = "Thomas J. Pickett and Ann Coleman", title = "Another continued fraction for {$ \pi $}", journal = j-AMER-MATH-MONTHLY, volume = "115", number = "10", pages = "930--933", year = "2008", CODEN = "AMMYAE", ISSN = "0002-9890 (print), 1930-0972 (electronic)", ISSN-L = "0002-9890", MRclass = "11A55", MRnumber = "2468553", bibdate = "Wed Jan 14 13:22:34 2015", bibsource = "http://www.math.utah.edu/pub/tex/bib/pi.bib", URL = "http://www.jstor.org/stable/27642639", acknowledgement = ack-nhfb, fjournal = "American Mathematical Monthly", journal-URL = "http://www.jstor.org/page/journal/amermathmont/about.html", } @Article{Pilehrood:2008:ALC, author = "Kh. Hessami Pilehrood and T. Hessami Pilehrood", title = "An {Ap{\'e}ry}-like continued fraction for {$ \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 = "http://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 = "http://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 = "http://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, libnote = "Not in my library.", 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 = "http://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 = "http://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 = "http://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 = "http://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", } @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 = "http://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 = "http://www.math.utah.edu/pub/tex/bib/pi.bib; http://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; http://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, $\delta(x) = (1 / 2 \pi) \int_{-\infty}^\infty e^{-ikx}\,dk$, namely, $\delta(x) = [(2 - q) / 2 \pi] \int_{-\infty}^{\infty} e_q^{-ikx}\,dk$, using the non-extensive-statistical-mechanics $q$-exponential function, $e_q^{ix} \equiv [1 + (1 - q) ix]^{1/(1 - q)}$ 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; http://www.math.utah.edu/pub/tex/bib/amermathmonthly2010.bib; http://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 = "http://www.jstor.org/page/journal/amermathmont/about.html", } @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 = "http://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 = "http://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; http://www.math.utah.edu/pub/tex/bib/amermathmonthly2010.bib; http://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 = "http://www.jstor.org/page/journal/amermathmont/about.html", } @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 = "http://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 = "http://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 = "http://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 = "http://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 = "http://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 = "http://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 = "http://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 = "http://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 = "http://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, } @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 = "http://www.math.utah.edu/pub/tex/bib/pi.bib", URL = "http://crd.lbl.gov/~dhbailey/dhbpapers/bbp-bluegene.pdf", acknowledgement = ack-nhfb, 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 = "http://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 $_3F_2({a, b, c} \atop {d, e} | 1)$ functions. We likewise prove and extend a corresponding integral identity given in \cite{Jauregui:2010:NRD}.", acknowledgement = ack-nhfb, } @Article{Chu:2011:DBS, author = "Wenchang Chu", title = "{Dougall}'s bilateral {$_2H_2$-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; http://www.math.utah.edu/pub/tex/bib/mathcomp2010.bib; http://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 $_2H_2$-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 = "http://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 = "http://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 = "http://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 = "http://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 = "http://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 = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://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/; http://www.math.utah.edu/pub/tex/bib/jmathphys2010.bib; http://www.math.utah.edu/pub/tex/bib/pi.bib", URL = "http://jmp.aip.org/resource/1/jmapaq/v53/i7/p073708_s1", acknowledgement = ack-nhfb, fjournal = "Journal of Mathematical Physics", journal-URL = "http://jmp.aip.org/", onlinedate = "18 July 2012", } @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 = "http://www.math.utah.edu/pub/tex/bib/expmath.bib; http://www.math.utah.edu/pub/tex/bib/pi.bib", acknowledgement = ack-nhfb, fjournal = "Experimental Mathematics", journal-URL = "http://www.tandfonline.com/loi/uexm20", } @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 = "http://www.math.utah.edu/pub/tex/bib/applmathcomput2010.bib; http://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, fjournal = "Applied Mathematics and Computation", journal-URL = "http://www.sciencedirect.com/science/journal/00963003", } @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 = "http://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; http://www.math.utah.edu/pub/tex/bib/numana2010.bib; http://www.math.utah.edu/pub/tex/bib/numeralgorithms.bib; http://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-7 = "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 = "http://www.math.utah.edu/pub/tex/bib/ieeeannhistcomput.bib; http://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 = "http://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 $", crossref = "Alladi:2013:RPW", pages = "103--109", year = "2013", DOI = "https://doi.org/10.1007/978-81-322-0767-2_16", bibdate = "Sat Sep 03 17:04:24 2016", bibsource = "http://www.math.utah.edu/pub/tex/bib/pi.bib", note = "This article appeared in \booktitle{The Hindu}, India s national newspaper in December 1994 on Ramanujan's 107th birth anniversary.", URL = "http://link.springer.com/chapter/10.1007/978-81-322-0767-2_16", acknowledgement = ack-nhfb, } @Article{AragonArtacho:2013:WRN, author = "Francisco {Arag{\'o}n Artacho} and David H. Bailey and Jonathan M. Borwein and Peter B. Borwein", title = "Walking on Real Numbers", journal = j-MATH-INTEL, volume = "35", number = "1", pages = "42--60", month = mar, year = "2013", CODEN = "MAINDC", DOI = "https://doi.org/10.1007/s00283-012-9340-x", ISSN = "0343-6993 (print), 1866-7414 (electronic)", ISSN-L = "0343-6993", bibdate = "Fri Mar 15 11:52:16 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/pi.bib; http://www.math.utah.edu/pub/tex/bib/prng.bib", URL = "http://gigapan.com/gigapans/106803; http://www.davidhbailey.com/dhbpapers/tools-walk.pdf", acknowledgement = ack-nhfb, fjournal = "The Mathematical Intelligencer", keywords = "Catalan's constant; Champernowne numbers; continued fractions; Copeland--Erd{\H{o}}s numbers; DNA genome numbers; dragon curves; Erd{\H{o}}s--Borwein numbers; Euler--Mascherino constant ($\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", } @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 = "http://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, } @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 = "http://www.math.utah.edu/pub/tex/bib/pi.bib", URL = "http://www.carma.newcastle.edu.au/jon/pi-monthly.pdf", acknowledgement = ack-nhfb, } @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; http://www.math.utah.edu/pub/tex/bib/compphyscomm2010.bib; http://www.math.utah.edu/pub/tex/bib/computing.bib; http://www.math.utah.edu/pub/tex/bib/pi.bib; http://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", } @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 = "http://www.math.utah.edu/pub/tex/bib/compphyscomm2010.bib; http://www.math.utah.edu/pub/tex/bib/pi.bib; http://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", } @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; http://www.math.utah.edu/pub/tex/bib/amermathmonthly2010.bib; http://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 = "http://www.jstor.org/page/journal/amermathmont/about.html", } @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 = "http://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 = "http://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; http://www.math.utah.edu/pub/tex/bib/amermathmonthly2010.bib; http://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 = "http://www.jstor.org/page/journal/amermathmont/about.html", } @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/; http://www.math.utah.edu/pub/tex/bib/pi.bib; http://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 = "http://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; http://www.math.utah.edu/pub/tex/bib/amermathmonthly2010.bib; http://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, fjournal = "American Mathematical Monthly", journal-URL = "http://www.jstor.org/page/journal/amermathmont/about.html", } @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 = "http://www.math.utah.edu/pub/tex/bib/pi.bib", acknowledgement = ack-nhfb, } @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 = "http://www.math.utah.edu/pub/bibnet/authors/b/borwein-jonathan-m.bib; http://www.math.utah.edu/pub/tex/bib/expmath.bib; http://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 = "http://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; http://www.math.utah.edu/pub/tex/bib/pi.bib; http://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 = "http://www.math.utah.edu/pub/tex/bib/amermathmonthly2010.bib; http://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, fjournal = "American Mathematical Monthly", journal-URL = "http://www.jstor.org/journals/00029890.html", } @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; http://www.math.utah.edu/pub/tex/bib/amermathmonthly2010.bib; http://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, 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 = "http://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 = "http://www.math.utah.edu/pub/tex/bib/jmathphys2015.bib; http://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 = "http://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 = "http://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 = "http://www.math.utah.edu/pub/tex/bib/histmath.bib; http://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 = "http://www.math.utah.edu/pub/bibnet/authors/b/borwein-jonathan-m.bib; http://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, fjournal = "Experimental Mathematics", journal-URL = "http://www.tandfonline.com/loi/uexm20", onlinedate = "24 August 2016", 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 = "http://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 = "http://www.math.utah.edu/pub/tex/bib/pi.bib", URL = "http://mathscholar.org/pi-and-the-collapse-of-peer-review", acknowledgement = ack-nhfb, 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 = "http://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 = "http://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 = "http://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 = "http://www.math.utah.edu/pub/tex/bib/agm.bib; http://www.math.utah.edu/pub/tex/bib/jautomreason.bib; http://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 = "http://www.math.utah.edu/pub/tex/bib/parallelcomputing.bib; http://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", }

%%% ==================================================================== %%% 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 = "http://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; http://www.math.utah.edu/pub/bibnet/authors/b/borwein-jonathan-m.bib; http://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 = "http://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 = "http://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{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 = "http://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 = "http://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", ISBN = "3-642-36735-6, 3-642-36736-4", DOI = "https://doi.org/10.1007/978-3-642-36736-6", 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; http://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; http://www.math.utah.edu/pub/tex/bib/einstein.bib; http://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", }