Notable Properties of Specific Numbers
First page . . . Back to page 24
 William Williams, Primitive history from the Creation to Cadmus. (1789). On page 4 we find:
[...] Next, to correct Meto's cycle answerably, [...] 334 years: which 121,991 days exceed by 90 minutes; and 334 tropical years exceed 4131 lunations just as much.
 John Narrien, An historical account of the origin and progress of astronomy. (1833). In chapter XI, page 232 we find:
[...] from the same authority we learn that Hippaichus had discovered, by a comparison of eclipses in whnch the moon's anomaly and latitude were the same, that in 5458 months, or 161,178 days, there were 5923 restitutions of latitude.
 T. J. J. See, Note on the accuracy of the Gaussian constant of the Solar system, Astronomische Nachrichten 166 89 (1904).
 W.W. Rouse Ball, "Four Fours. Some Arithmetical Puzzles.", in The Mathematical Gazette, 6(98), May 1912.
 Godfrey H. Hardy, Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, Cambridge, 1940 (also republished in 1959, 1978, and 1999).
 Kasner and Newman, Mathematics and the Imagination, (Simon and Schuster, New York) 1940 (also republished in 1989 and in 2001). The story can also be found online — search for Googol plus the leading sentence "Words of wisdom are spoken by children at least as often as by scientists."
 Claude Shannon, Prediction and Entropy of Printed English, Bell System Technical Journal 28 pp. 50-64 (1951).
 T. Nagell, The diophantine equation x2+7=2n. Archiv fur Mathematik 4(13) pp. 185-187 (1960). Available from Springer
 S. Knapowski, On sign-changes of the difference π(x) - li x. Acta Arithmetica 7, 107-119 (1962).
 Dmitri Borgmann, "Naming the Numbers", Word Ways: the Journal of Recreational Linguistics 1 (1), February 1968. Cover and contents are here and article is here.
 V. E. Hoggat Jr. and C. T. Long, Divisibility properties of generalised fibonacci polynomials, Fibonacci Quarterly 113 (1973).
 Dennis Ritchie, Fifth Edition UNIX (Bell Laboratories), sqrt.s (PDP-11 assembler source code for a C library routine), June 1974. Archive created by The Unix Heritage Society. (I first found this on a mirror here).
 Nancy Bowers and Pundia Lepi, Kaugel Valley systems of reckoning, Journal of the Polynesian Society 84 (3), pp. 309-324.
 David Singmaster, Repeated Binomial Coefficients and Fibonacci Numbers, Fibonacci Quarterly 13 (1975), pp. 295-298.
 Ted Bastin et al., On the physical interpretation and the mathematical structure of the combinatorial hierarchy, Int. J. Theor. Phys. 18 p. 445 (1979). PDF here.
 David A. Klarner, The Mathematical Gardner, 1980. ISBN 0-534-98015-5
 Donald E. Knuth and Allan A. Miller, "A Programming and Problem-Solving Seminar" (notes from Stanford CS 204, Fall 1980), pages 4-12. PDF here: Programming and Problem-Solving Seminar
 Carl Sagan, Ann Druyan and Steven Soter (creators), Cosmos: a Personal Voyage (television series), 1980. Episode 9 has the googol quote.
 Morwen B. Thistlethwaite, untitled (cover letter and computer listings) (describing a "52-move strategy for solving Rubik's Cube"), 1981.
 J. H. Conway and N. J. A. Sloane, Lorentzian forms for the Leech lattice. Bulletin of the American Mathematical Society 6(2) (March 1982), pp. 215-217.
 T. Padmanabhan, "Inflation from quantum gravity", Phys. Letts., (1984), A104, pp 196-199.
 John Horton Conway and Richard Guy, The Book of Numbers, New York: Springer-Verlag (1996). ISBN 038797993X.
 Wells, David, The Penguin Dictionary of Curious and Interesting Numbers. (Original edition 1986; revised and expanded 1998).
 Richard Guy, The strong law of small numbers. The American Mathematical Monthly 95(8) pp. 697-712 (1988). This has been used for several university courses and when I last checked was available here, here and here. (also formerly at http://ndikandi.utm.mx/~lm2002070425/Guy.pdf)
 The Compact Oxford English Dictionary (Second Edition), 1991. This is the version that has 21473 pages photographically reduced into a single book of about 2400 pages.
 J. Meeus and D. Savoie, The history of the tropical year, Journal of the British Astronomical Association 102(1) pp. 40-42 (1992)
 Linda Scele Drawings Collection, Scele drawing 4087, 1993.
 Don N. Page, Information loss in black holes and/or conscious beings? , 1994. arXiv:hep-th/9411193v2
 Simon et al., Numerical expressions for precession formulae and mean elements for the Moon and the planets (1994).
 James G. Gilson, Calculating the fine structure constant, 1995. PDF here
 Maurice Mignotte and Attila Pethö, On the system of diophantine equations x2-6y2=-5 and x=2z2-1. Mathematica Scandinavica 76, pp. 50-60 (1995). Available from the publisher here
 H. Pierre Noyes. Measurement, accuracy, bit-strings, Manthey's quaternions, and RRQM. In Entelechies (Proc. ANPA 16), K. G. Bowden, ed., University of East London. pp. 27-50. PDF here
 H. Pierre Noyes. Some remarks on discrete physics as an ultimate dynamical theory. PDF here
 David Singmaster, Chronology of Recreational Mathematics, 1996.
 Jim Blinn, Floating-point tricks, IEEE Computer Graphics and Applications, 1997.
 Richard Crandall, "The Challenge of Large Numbers", Scientific American no. 276 (Feb. 1997), pp. 74-79.
 D. E. Knuth. The Art of Computer Programming. vol 4A. Combinatorial Algorithms.
 J. H. E. Cohn, The diophantine system x2-6y2=-5, x=2z2-1. Mathematica Scandinavica 82, pp. 161-164 (1998). Available from the publisher here
 Eric Weisstein, The CRC Concise Encyclopedia of Mathematics (CRC Press), 1998. ISBN 0849396409.
 Richard Borcherds, The Leech lattice and other lattices. Ph.D. thesis, Trinity College (originally given June 1984), as corrected in 1999.
 Georges Ifrah, The Universal History of Numbers, ISBN 0-471-37568-3. (1999).
 Patrick Costello, A New Largest Smith Number, Fibonacci Quarterly 40(4) 369-371, 2002.
 Erich Friedman, Problem of the month (August 2000), web page, 2000-2009.
 Erich Friedman, What's Special About This Number?, web page, 2000-2009.
 John Baez, The Fano Plane (web page) 2001. (Part of a collection describing the Octonions)
 Palais, Robert. "π Is Wrong!". The Mathematical Intelligencer 23 (3) 7–8 (2001).
 David Eberly, Fast inverse square root, 2002 (as archived on 2003 Apr 26 by the Internet Archive Wayback Machine).
 Michael Janssen, The Trouton experiment and E=mc2 (handout, PDF file), 2002.
 Eric Balandraud, Calculating the Permutations of 4D Magic Cubes, 2003.
 Toshio Fukushima, A new precession formula. Public Relations Center, NAOJ, 2003.
 Chris Lomont, Fast inverse square root, 2003.
 Byron Schmuland, "Shouting Factorials!", 23 Oct 2003.
 Max Tegmark, Parallel Universes, 2003. Available from arxiv.org.
 M. Agrawal et al., PRIMES is in P. Annals of Mathematics 160(2) pp. 781-793 (2004). Available from the editors here.
 Dario Alpern, Known 3-digit prime factors of Googolplexplex - 1, web site, 2004. http://www.alpertron.com.ar/glpxm1.pl?digits=3
 Sean M. Carroll and Jennifer Chen, Spontaneous Inflation and the Origin of the Arrow of Time, PDF on arXiv
 Tamara M. Davis and Charles H. Lineweaver, Expanding Confusion: Common Misconceptions of Cosmological Horizons and the Superluminal Expansion of the Universe, 2004.
 Maohua Le, On the diophantine system x2-Dy2=1-D AND x=2z2-1. Mathematica Scandinavica 95, pp. 171-180 (2004). Available from the publisher here.
 Sergey N. Dorogovtsev, José Fernando F. Mendes, and Joao Gama Oliveira. "Frequency of occurrence of numbers in the World Wide Web." Physica A: Statistical Mechanics and its Applications 360.2 (2006): 548-556. on arXiv here
 Gordon, Raymond G., Jr. (ed.), Ethnologue: Languages of the World (15th edition), SIL International, Dallas (2005). Online version at www.ethnologue.com
 http://space.mit.edu/~kcooksey/teaching/AY5/MisconceptionsabouttheBigBang_ScientificAmerican.pdf Charles H. Lineweaver and Tamara M. Davis, Misconceptions about the Big Bang, Scientific American, February 2005.
 Clifford Pickover. A Passion for Mathematics: numbers, puzzles, madness, religion, and the quest for reality. Wiley (2005). ISBN 0-471-69098-8.
 John Baez, Klein's Quartic Curve (web page) July 28, 2006.
 Bailey, Borwein, Kapoor and Weisstein, Ten Problems in Experimental Mathematics, American Mathematical Monthly, 2006.
 Andrew Granville and Greg Martin, Prime number races, The American Mathematical Monthly 113(1) pp. 1-33 (2006). Available from the AMM here; a 2004 preprint is on arxiv.org.
 Don N. Page, Susskind's challenge to the Hartle-Hawking no-boundary proposal and possible resolutions, 2006. arXiv:hep-th/0610199v2
 Mark Ronan, Symmetry and the Monster: The Story of One of the Greatest Quests of Mathematics, 2006. ISBN 0-19-280723-4
 Alan H. Guth, Eternal inflation and its implications, 2nd International Conference on Quantum Theories and Renormalization Group in Gravity and Cosmology (IRGAC2006), Barcelona, Spain, 11-15 July 2006.
 David de Neufville, personal correspondence.
 John Baez, My Favorite Numbers (web page) 2008. Includes videos and slides from three talks given in 2008 at University of Glasgow.
 Andrew Granville, Prime number patterns, The American Mathematical Monthly 115(4) pp. 279-296 (2008). Available from the MAA here and from the author here.
 My Math Forum, discussion thread, 2008 Oct 10
 N. J. A. Sloane, Eight Hateful Sequences, 2008.
 Ken Auletta, Googled : the end of the world as we know it (New York : Penguin Press, 2009) ISBN 9781594202353.
 Daan van Berkel, On a curious property of 3435. (2009) arXiv:0911.3038
 CNN Beat 360, Anderson Cooper Daily Podcast for July 15th, 2009.
 Huffington Post, Man Charged 23 Quadrillion..., July 15th, 2009.
 Andrei Linde and Vitaly Vanchurin, How many universes are in the multiverse?, 2009. arXiv:0910.1589v2
 WMUR TV-9 (Manchester NH), Man's Debit Card Charged $23 Quadrillion..., July 15th, 2009.
 http://www.astro.ucla.edu/~wright/cosmology_faq.html Edward L. Wright, Frequently Asked Questions in Cosmology (web page), 2009.
 WTOV, Card Users Hit With $23 Quadrillion Charge, July 15th, 2009.
 David Eberly, Fast inverse square root (revisited), 2010.
 Jeffrey Hankins, personal correspondence, 2010.
 Theodore P. Hill, Ronald F. Fox, Jack Miller, A Better Definition of the Kilogram
(note on page 5: "At this point in time, it is not yet possible to obtain exact counts of individual atoms, even when they are in a crystal lattice, but that is merely a question of time.")
 David Stuart, Notes on Accession Dates in the Inscriptions of Coba, 2010. Available here.
 Mark R. Diamond, Multiplicative persistence base 10: some new null results, 2011.
 Nicolas Gauvit et al., Sloane's Gap: Do Mathematical and Social Factors Explain the Distribution of Numbers in the OEIS?, 2011.
 Ivan Panchenko, personal correspondence, 2011.
 Marek Wolf, The "Skewes' number" for twin primes: counting sign changes of π2(x) − C2Li2(x), 2011. Available from arxiv.org.
 Bob Delaney, "fp Plugin 5.1", message to realbasic-nug forum (mirror here), 30 Jan 2012
 Adam Goucher, Lunisolar calendars (blog article), 2012.
 Gottfried Helms, The Lucas-Lehmer-test for Mersenne-numbers and the number Λ ~1.389910663524..., April 4 2012.
 Randall Munroe, xkcd 1047 -- Approximations (online comic strip), April 25 2012.
Note : This strip mentions my ries program because Munroe used it to derive some of the expressions, near-equalities and approximations shown in the strip. He and I did not communicate prior to the publication of the strip, and all of the material in the strip was found by him. Answering a presumably large volume of responses, he specifically commented on this fact in a note at the top of the comic (which was visible for a while on the first day) by stating:
"Note: '1 year = π × 107 seconds' is popular with physicists. For this list, I've tried to stick to approximations that I noticed on my own."
There are a few obvious exceptions which were included for their amusement value: the Rent approximation 525600×60 ≈ 31556952, and 1/140 as an approximation to the reciprocal of the fine-structure constant (the comment "I've had enough of this 137 crap" refers to the fanatical cult of 137).
 Robert Munafo, answer to a question by Mahmud. The relevant discussion is also here: What happens when numbers become large... really large?
 "Pat's Blog", Before there were four fours..., 2012.
 TrueNews.org, "The Origin of Life -- Evolution's Dilemma (web page), accessed 2010 April 29.
 Wolfram Alpha, "computational knowledge engine" online resource.
 27: This is close.
 Simon Singh, "The Simpsons and Their Mathematical Secrets" (book, 2013) Some of the material is also presented in this article by Singh for The Guardian.
 Inder J. Taneja, "Crazy Sequential Representation: Numbers from 0 to 11111 in terms of Increasing and Decreasing Orders of 1 to 9" (2014) on arxiv
 Washington Taylor and Yi-Nan Wang, "The F-theory geometry with most flux vacua". Journal of High Energy Physics. 2015(12): 164 (2015) on arXiv
 John Tromp, "Number of legal Go positions" (2016).
 James Read and Baptiste Le Bihan. "The landscape and the multiverse: What's the problem?". Synthese. 199(3–4) 7749–7771 (2021)
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