%%% -*-BibTeX-*-
%%% ====================================================================
%%%  BibTeX-file{
%%%     author          = "Nelson H. F. Beebe",
%%%     version         = "1.58",
%%%     date            = "18 March 2014",
%%%     time            = "17:34:40 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        = "32712 7081 31590 311395",
%%%     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 to the diameter of a circle.
%%%                        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 1.58, the year coverage looked
%%%                        like this:
%%%
%%%                             1727 (   1)    1823 (   0)    1919 (   0)
%%%                             1729 (   0)    1825 (   0)    1921 (   1)
%%%                             1732 (   0)    1828 (   0)    1924 (   1)
%%%                             1733 (   0)    1829 (   0)    1925 (   1)
%%%                             1734 (   0)    1830 (   0)    1926 (   2)
%%%                             1738 (   0)    1834 (   0)    1930 (   1)
%%%                             1739 (   0)    1835 (   0)    1931 (   1)
%%%                             1741 (   0)    1837 (   0)    1933 (   1)
%%%                             1743 (   0)    1839 (   0)    1935 (   1)
%%%                             1746 (   0)    1842 (   0)    1938 (   2)
%%%                             1747 (   0)    1843 (   0)    1939 (   2)
%%%                             1748 (   0)    1844 (   0)    1940 (   1)
%%%                             1750 (   0)    1846 (   0)    1942 (   1)
%%%                             1753 (   0)    1849 (   0)    1945 (   1)
%%%                             1754 (   0)    1850 (   0)    1946 (   2)
%%%                             1755 (   0)    1851 (   0)    1947 (   2)
%%%                             1756 (   0)    1852 (   0)    1948 (   1)
%%%                             1757 (   0)    1853 (   1)    1949 (   0)
%%%                             1758 (   0)    1854 (   0)    1950 (   3)
%%%                             1762 (   0)    1858 (   0)    1954 (   1)
%%%                             1763 (   0)    1859 (   0)    1955 (   4)
%%%                             1765 (   0)    1861 (   0)    1957 (   1)
%%%                             1766 (   0)    1862 (   0)    1958 (   1)
%%%                             1767 (   0)    1863 (   0)    1959 (   1)
%%%                             1768 (   1)    1864 (   0)    1960 (   2)
%%%                             1769 (   0)    1865 (   0)    1961 (   1)
%%%                             1770 (   0)    1866 (   0)    1962 (   3)
%%%                             1773 (   0)    1869 (   0)    1965 (   1)
%%%                             1775 (   0)    1871 (   2)    1967 (   3)
%%%                             1776 (   0)    1872 (   0)    1968 (   1)
%%%                             1777 (   0)    1873 (   1)    1969 (   3)
%%%                             1778 (   0)    1874 (   0)    1970 (   2)
%%%                             1779 (   0)    1875 (   0)    1971 (   2)
%%%                             1780 (   0)    1876 (   0)    1972 (   1)
%%%                             1781 (   0)    1877 (   0)    1973 (   1)
%%%                             1783 (   0)    1879 (   1)    1975 (   0)
%%%                             1784 (   0)    1880 (   0)    1976 (   4)
%%%                             1785 (   0)    1881 (   0)    1977 (   1)
%%%                             1786 (   0)    1882 (   1)    1978 (   2)
%%%                             1787 (   0)    1883 (   1)    1979 (   3)
%%%                             1788 (   0)    1884 (   0)    1980 (   2)
%%%                             1789 (   0)    1885 (   0)    1981 (   3)
%%%                             1790 (   0)    1886 (   0)    1982 (   1)
%%%                             1791 (   0)    1887 (   0)    1983 (   3)
%%%                             1792 (   0)    1888 (   0)    1984 (   2)
%%%                             1793 (   0)    1889 (   0)    1985 (   2)
%%%                             1794 (   0)    1890 (   0)    1986 (   7)
%%%                             1795 (   0)    1891 (   1)    1987 (   4)
%%%                             1796 (   0)    1892 (   0)    1988 (   8)
%%%                             1797 (   0)    1893 (   0)    1989 (   6)
%%%                             1798 (   0)    1894 (   0)    1990 (   3)
%%%                             1799 (   0)    1895 (   1)    1991 (   3)
%%%                             1800 (   0)    1896 (   1)    1992 (   3)
%%%                             1801 (   0)    1897 (   0)    1993 (   4)
%%%                             1802 (   0)    1898 (   0)    1994 (   4)
%%%                             1803 (   0)    1899 (   0)    1995 (   4)
%%%                             1804 (   0)    1900 (   0)    1996 (   6)
%%%                             1805 (   0)    1901 (   0)    1997 (  12)
%%%                             1806 (   0)    1902 (   0)    1998 (   6)
%%%                             1807 (   0)    1903 (   0)    1999 (   4)
%%%                             1808 (   0)    1904 (   1)    2000 (   7)
%%%                             1809 (   0)    1905 (   0)    2001 (   6)
%%%                             1810 (   0)    1906 (   0)    2002 (   3)
%%%                             1811 (   0)    1907 (   0)    2003 (   5)
%%%                             1812 (   0)    1908 (   0)    2004 (   6)
%%%                             1813 (   0)    1909 (   0)    2005 (   5)
%%%                             1814 (   0)    1910 (   0)    2006 (   4)
%%%                             1815 (   0)    1911 (   0)    2007 (   1)
%%%                             1816 (   0)    1912 (   0)    2008 (   7)
%%%                             1817 (   0)    1913 (   0)    2009 (   2)
%%%                             1818 (   0)    1914 (   0)    2010 (  10)
%%%                             1819 (   0)    1915 (   0)    2011 (  15)
%%%                             1820 (   0)    1916 (   0)    2012 (   6)
%%%                             1821 (   0)    1917 (   0)    2013 (  12)
%%%                             1822 (   0)    1918 (   0)    2014 (   4)
%%%
%%%                             Article:        183
%%%                             Book:            21
%%%                             InBook:           1
%%%                             InCollection:     3
%%%                             InProceedings:    9
%%%                             Misc:             5
%%%                             Proceedings:      3
%%%                             TechReport:      11
%%%                             Unpublished:     19
%%%
%%%                             Total entries:  255
%%%
%%%                        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.
%%%
%%%                        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''.
%%%
%%%                        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 \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-MATH-APPL        = "Computers and Mathematics with Applications"}

@String{j-COMP-PHYS-COMM        = "Computer Physics Communications"}

@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-J-ALG                 = "Journal of Algorithms"}

@String{j-J-ACM                 = "Journal of the ACM"}

@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-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-PAC-J-MATH            = "Pacific Journal of Mathematics"}

@String{j-PARALLEL-COMPUTING    = "Parallel Computing"}

@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-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"}

%%% ====================================================================
%%% 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-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-PROMETHEUS-BOOKS    = "Prometheus Books"}
@String{pub-PROMETHEUS-BOOKS:adr = "Amherst, NY, 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."}

%%% ====================================================================
%%% Series abbreviations:

@String{ser-LNCS                = "Lecture Notes in Computer Science"}

%%% ====================================================================
%%% Bibliography entries, sorted by ascending year, and then by citation
%%% label, with ``bibsort --byyear'':

@String{j-INT-J-MOD-PHYS-C      = "International Journal of Modern Physics C [Physics and Computers]"}
@String{j-TRANS-INFO-PROCESSING-SOC-JAPAN = "Transactions of the Information Processing Society of Japan"}
@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{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 =          "http://dx.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",
}

@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 =          "http://dx.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{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{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:PS}.",
  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",
}

@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 =          "http://dx.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 =          "http://dx.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 =          "http://dx.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/pi.bib; JSTOR
                 database",
  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 =          "http://doi.acm.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",
}

@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 =          "http://doi.acm.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",
}

@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 =          "http://doi.acm.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 =          "http://dx.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 =          "http://dx.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 =          "http://doi.acm.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 =          "http://dx.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",
}

@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 =          "http://dx.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 =          "http://dx.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/",
}

@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/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 =          "http://dx.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{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 =          "http://www.springerlink.com/openurl.asp?genre=article&id=doi: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",
  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 =          "http://dx.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 =          "http://dx.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://ieeexplore.ieee.org/;
                 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",
  ISSN =         "0002-9890 (print), 1930-0972 (electronic)",
  ISSN-L =       "0002-9890",
  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 =          "http://dx.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 =          "http://www.springerlink.com/openurl.asp?genre=article&id=doi: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{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 =          "http://dx.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/pi.bib;
                 z3950.loc.gov:7090/Voyager",
  price =        "US\$30.00 (Oxford Univ. Press)",
  acknowledgement = ack-nhfb,
  libnote =      "Not in my library.",
  subject =      "Mathematics",
}

@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,
}

@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 =          "http://dx.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 =          "http://doi.acm.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 $\pi$ [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/pi.bib",
  note =         "Reprint of the third edition of 1971.",
  price =        "US\$6.98",
  acknowledgement = ack-nhfb,
  xxnote =       "Fourth edition, 1977, Golem Press, Boulder, CO, ISBN
                 0-911762-18-3, LCCN QA484 .B4 1977, also available.",
}

@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",
}

@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 =          "http://doi.acm.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/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/pi.bib;
                 z3950.loc.gov:7090/Voyager",
  acknowledgement = ack-nhfb,
  libnote =      "Not in my library.",
  remark =       "Originally published: Cambridge: Oxford University,
                 1992.",
  subject =      "Mathematics",
}

@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 =     "$\pi$",
}

@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:PS,
  editor =       "Lennart Berggren and Jonathan Borwein and Peter
                 Borwein",
  title =        "Pi, a sourcebook",
  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 =      "Sat Apr 23 09:55:09 MDT 2011",
  bibsource =    "http://www.math.utah.edu/pub/tex/bib/pi.bib;
                 melvyl.cdlib.org:210/CDL90",
  acknowledgement = ack-nhfb,
  subject =      "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/pi.bib;
                 z3950.loc.gov:7090/Voyager",
  URL =          "http://www.walkerbooks.com/books/catalog.php?key=4",
  acknowledgement = ack-nhfb,
  libnote =      "Not in my library.",
  subject =      "Pi (mathematical constant)",
}

@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 =          "http://dx.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 =          "http://dx.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 =          "http://doi.acm.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,
}

@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",
  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",
}

@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{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 =          "http://dx.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.computer.org/cse/cs1999;
                 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:PS,
  editor =       "Lennart Berggren and Jonathan Borwein and Peter
                 Borwein",
  title =        "{Pi}, a sourcebook",
  publisher =    pub-SV,
  address =      pub-SV:adr,
  edition =      "Second",
  pages =        "xix + 736",
  year =         "2000",
  ISBN =         "0-387-98946-3 (hardcover)",
  ISBN-13 =      "978-0-387-98946-4 (hardcover)",
  LCCN =         "QA484 .P5 2000",
  bibdate =      "Mon Aug 10 17:48:29 MDT 2009",
  bibsource =    "http://www.math.utah.edu/pub/tex/bib/pi.bib;
                 z3950.loc.gov:7090/Voyager",
  acknowledgement = ack-nhfb,
  libnote =      "Not yet in my library.",
  subject =      "Pi (mathematical constant)",
}

@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).",
}

@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",
  ISBN-13 =      "978-3-540-66572-4",
  LCCN =         "QA484.A7513 2001",
  bibdate =      "Sat Apr 20 11:01:28 2002",
  bibsource =    "http://www.math.utah.edu/pub/tex/bib/pi.bib",
  note =         "Includes CD-ROM. Translated from the German by
                 Catriona and David Lischka.",
  acknowledgement = ack-nhfb,
}

@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{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{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",
  ISBN-13 =      "978-0-521-81805-6",
  LCCN =         "QA41 .F54 2003",
  bibdate =      "Mon Dec 31 07:47:16 MST 2007",
  bibsource =    "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",
  acknowledgement = ack-nhfb,
  libnote =      "Not in my library.",
  subject =      "Mathematical constants",
}

@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 =          "http://dx.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",
}

@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,
}

@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 =          "http://dx.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://csdl.computer.org/comp/mags/cs/2005/03/c3toc.htm;
                 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 =          "http://dx.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",
  month =        "????",
  year =         "2005",
  CODEN =        "JSCSAJ",
  DOI =          "http://dx.doi.org/10.1080/00949650410001687181",
  ISSN =         "0094-9655 (print), 1026-7778 (electronic), 1563-5163",
  ISSN-L =       "0094-9655",
  bibdate =      "Sat Jan 07 08:42:35 2012",
  bibsource =    "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",
  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",
}

@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{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{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 =          "http://dx.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.computer.org/annals/an2002/;
                 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{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 =          "http://dx.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 =          "http://dx.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 =          "http://dx.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-COMP-MATH-APPL,
  volume =       "56",
  number =       "10",
  pages =        "2772--2778",
  month =        nov,
  year =         "2008",
  CODEN =        "CMAPDK",
  DOI =          "http://dx.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",
}

@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",
}

@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 =          "http://dx.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",
}

@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{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 =          "http://dx.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 =          "http://dx.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 =          "http://dx.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{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{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 =          "http://dx.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 =          "http://dx.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 =          "http://dx.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/",
}

@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 =          "http://dx.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 =          "http://dx.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{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 =          "http://dx.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%2Fs00283-012-9312-1",
  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 =          "http://dx.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 =          "http://dx.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 =          "http://dx.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/",
}

@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 =          "http://dx.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 =          "http://dx.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 =          "http://dx.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 =          "http://dx.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 =          "http://dx.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 =          "http://dx.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 =          "http://dx.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{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",
}

%%% ====================================================================
%%% 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{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 =          "http://dx.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{Sidoli:2014:ATB,
  editor =       "Nathan Sidoli and Glen {Van Brummelen}",
  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 =          "http://dx.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.",
}