Notable Properties of Specific Numbers  

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[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] W.W. Rouse Ball, "Four Fours. Some Arithmetical Puzzles.", in The Mathematical Gazette, 6(98), May 1912.

[135] Godfrey H. Hardy, Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, Cambridge, 1940 (also republished in 1959, 1978, and 1999).

[136] 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."

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

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

[139] 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.

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

[141] 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).

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

[143] David Singmaster, Repeated Binomial Coefficients and Fibonacci Numbers, Fibonacci Quarterly 13 (1975), pp. 295-298.

[144] 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.

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

[146] 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

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

[148] Morwen B. Thistlethwaite, untitled (cover letter and computer listings) (describing a "52-move strategy for solving Rubik's Cube"), 1981.

[149] 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.

[150] T. Padmanabhan, "Inflation from quantum gravity", Phys. Letts., (1984), A104, pp 196-199.

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

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

[153] 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

[154] 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.

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

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

[157] Don N. Page, Information loss in black holes and/or conscious beings? , 1994. arXiv:hep-th/9411193v2

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

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

[160] 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

[161] 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

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

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

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

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

[166] 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

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

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

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

[170] Patrick Costello, A New Largest Smith Number, Fibonacci Quarterly 40(4) 369-371, 2002.

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

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

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

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

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

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

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

[178] Chris Lomont, Fast inverse square root, 2003.

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

[180] Max Tegmark, Parallel Universes, 2003. Available from

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

[182] Dario Alpern, Known 3-digit prime factors of Googolplexplex - 1, web site, 2004.

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

[184] 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.

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

[186] Clifford Pickover. A Passion for Mathematics: numbers, puzzles, madness, religion, and the quest for reality. Wiley (2005). ISBN 0-471-69098-8.

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

[188] Bailey, Borwein, Kapoor and Weisstein, Ten Problems in Experimental Mathematics, American Mathematical Monthly, 2006.

[189] 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

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

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

[192] 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.

[193] David de Neufville, personal correspondence.

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

[195] 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.

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

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

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

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

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

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

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

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

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

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

[206] Jeffrey Hankins, personal correspondence, 2010.

[207] 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.")

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

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

[210] Nicolas Gauvit et al., Sloane's Gap: Do Mathematical and Social Factors Explain the Distribution of Numbers in the OEIS?, 2011.

[211] Ivan Panchenko, personal correspondence, 2011.

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

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

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

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

[216] 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 525600×6031556952, 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).

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

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

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

[220] 27: This is close.

[221] Simon Singh, "The Simpsons and their Mathematical Secrets" (2014)

[222] 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

[223] John Tromp, "Number of legal Go positions" (2016).

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    109152051    101036    101010100    — —    footnotes    Also, check out my large numbers and integer sequences pages.

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