Notable Properties of Specific Numbers

Bibliography

[131] 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.

[132] 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.

[133] T. J. J. See, Note on the accuracy of the Gaussian constant of the Solar system, Astronomische Nachrichten 166 89 (1904).

[134] 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."

[135] T. Nagell, The diophantine equation x²+7=2ⁿ. Archiv fur Mathematik 4(13) pp. 185-187 (1960). Available from Springer

[136] S. Knapowski, On sign-changes of the difference π(x) - li x. Acta Arithmetica 7, 107-119 (1962).

[137] 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.

[138] V. E. Hoggat Jr. and C. T. Long, Divisibility properties of generalized fibonacci polynomials, Fibonacci Quarterly 113 (1973).

[139] 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).

[140] Nancy Bowers and Pundia Lepi, Kaugel Valley systems of reckoning, Journal of the Polynesian Society 84 (3), pp. 309-324.

[141] 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.

[142] David A. Klarner, The Mathematical Gardner, 1980. ISBN 0-534-98015-5

[143] 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

[144] Carl Sagan, Ann Druyan and Steven Soter (creators), Cosmos: a Personal Voyage (television series), 1980. Episode 9 has the googol quote.

[145] 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.

[146] John Horton Conway and Richard Guy, The Book of Numbers, New York: Springer-Verlag (1996). ISBN 038797993X.

[147] Wells, David, The Penguin Dictionary of Curious and Interesting Numbers. (Original edition 1986; revised and expanded 1998).

[148] 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)

[149] 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.

[150] J. Meeus and D. Savoie, The history of the tropical year, Journal of the British Astronomical Association 102(1) pp. 40-42 (1992)

[151] Linda Scele Drawings Collection, Scele drawing 4087, 1993.

[152] Simon et al., Numerical expressions for precession formulae and mean elements for the Moon and the planets (1994).

[153] James G. Gilson, Calculating the fine structure constant, 1995. PDF here

[154] 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

[155] 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

[156] H. Pierre Noyes. Some remarks on discrete physics as an ultimate dynamical theory. PDF here

[157] Jim Blinn, Floating-point tricks, IEEE Computer Graphics and Applications, 1997.

[158] Richard Crandall, "The Challenge of Large Numbers", Scientific American no. 276 (Feb. 1997), pp. 74-79.

[159] D. E. Knuth. The Art of Computer Programming. vol 4A. Combinatorial Algorithms.

[160] 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

[161] Eric Weisstein, The CRC Concise Encyclopedia of Mathematics (CRC Press), 1998. ISBN 0849396409.

[162] Richard Borcherds, The Leech lattice and other lattices. Ph.D. thesis, Trinity College (originally given June 1984), as corrected in 1999.

[163] Georges Ifrah, The Universal History of Numbers, ISBN 0-471-37568-3. (1999).

[164] Erich Friedman, Problem of the month (August 2000), web page, 2000-2009.

[165] Erich Friedman, What's Special About This Number?, web page, 2000-2009.

[166] John Baez, The Fano Plane (web page) 2001. (Part of a collection describing the Octonions)

[167] Palais, Robert. "π Is Wrong!". The Mathematical Intelligencer 23 (3) 7–8 (2001).

[168] David Eberly, Fast inverse square root, 2002 (as archived on 2003 Apr 26 by the Internet Archive Wayback Machine).

[169] Michael Janssen, The Trouton experiment and E=mc² (handout, PDF file), 2002.

[170] Toshio Fukushima, A new precession formula. Public Relations Center, NAOJ, 2003.

[171] Byron Schmuland, "Shouting Factorials!", 23 Oct 2003.

[172] Max Tegmark, Parallel Universes, 2003. Available from arxiv.org.

[173] M. Agrawal et al., PRIMES is in P. Annals of Mathematics 160(2) pp. 781-793 (2004). Available from the editors here.

[174] Dario Alpern, Known 3-digit prime factors of Googolplexplex - 1, web site, 2004. http://www.alpertron.com.ar/glpxm1.pl?digits=3

[175] Tamara M. Davis and Charles H. Lineweaver, Expanding Confusion: Common Misconceptions of Cosmological Horizons and the Superluminal Expansion of the Universe, 2004.

[176] 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.

[177] Gordon, Raymond G., Jr. (ed.), Ethnologue: Languages of the World (15^th edition), SIL International, Dallas (2005). Online version at www.ethnologue.com

[178] John Baez, Klein's Quartic Curve (web page) July 28, 2006.

[179] 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.

[180] Don N. Page, Susskind's challenge to the Hartle-Hawking no-boundary proposal and possible resolutions, 2006. arXiv:hep-th/0610199v2

[181] Mark Ronan, Symmetry and the Monster: The Story of One of the Greatest Quests of Mathematics, 2006. ISBN 0-19-280723-4

[182] Alan H. Guth, Eternal inflation and its implications, 2^nd International Conference on Quantum Theories and Renormalization Group in Gravity and Cosmology (IRGAC2006), Barcelona, Spain, 11-15 July 2006.

[183] David de Neufville, personal correspondence.

[184] John Baez, My Favorite Numbers (web page) 2008. Includes videos and slides from three talks given in 2008 at University of Glasgow.

[185] 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.

[186] My Math Forum, discussion thread, 2008 Oct 10

[187] N. J. A. Sloane, Eight Hateful Sequences, 2008.

[188] Ken Auletta, Googled : the end of the world as we know it (New York : Penguin Press, 2009) ISBN 9781594202353.

[189] Daan van Berkel, On a curious property of 3435. (2009) arXiv:0911.3038

[190] CNN Beat 360, Anderson Cooper Daily Podcast for July 15^th, 2009.

[191] Huffington Post, Man Charged 23 Quadrillion..., July 15^th, 2009.

[192] Andrei Linde and Vitaly Vanchurin, How many universes are in the multiverse?, 2009. arXiv:0910.1589v2

[193] WMUR TV-9 (Manchester NH), Man's Debit Card Charged $23 Quadrillion..., July 15^th, 2009.

[194] WTOV, Card Users Hit With $23 Quadrillion Charge, July 15^th, 2009.

[195] David Eberly, Fast inverse square root (revisited), 2010.

[196] Jeffrey Hankins, personal correspondence, 2010.

[197] 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.")

[198] David Stuart, Notes on Accession Dates in the Inscriptions of Coba, 2010. Available here.

[199] Mark R. Diamond, Multiplicative persistence base 10: some new null results, 2011.

[200] Ivan Panchenko, personal correspondence, 2011.

[201] Marek Wolf, The Skewes number for twin primes: counting sign changes of π₂(x) − C₂Li₂(x), 2011. Available from arxiv.org.

[202] Bob Delaney, "fp Plugin 5.1", message to realbasic-nug forum (mirror here), 30 Jan 2012

[203] Adam Goucher, Lunisolar calendars (blog article), 2012.

[204] Gottfried Helms, The Lucas-Lehmer-test for Mersenne-numbers and the number Λ ~1.389910663524..., April 4 2012.

[205] 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).

[206] Robert Munafo, answer to a question by Mahmud. The relevant discussion is also here: What happens when numbers become large... really large?

[207] TrueNews.org, "The Origin of Life -- Evolution's Dilemma (web page), accessed 2010 April 29.

[208] Wolfram Alpha, "computational knowledge engine" online resource.

[209] 27: This is close.

Quick index: if you're looking for a specific number, start with whichever of these is closest: 0.065988... 1 1.618033... 3.141592... 4 12 16 21 24 29 39 46 52 64 68 89 107 137.03599... 158 231 256 365 616 714 1024 1729 4181 10080 45360 262144 1969920 73939133 4294967297 5×10¹¹ 10¹⁸ 5.4×10²⁷ 10⁴⁰ 5.21...×10⁷⁸ 1.29...×10⁸⁶⁵ 10⁴⁰⁰⁰⁰ 10^9152051 10^10³⁶ 10^{10^10¹⁰⁰} — — footnotes Also, check out my large numbers and integer sequences pages.