Notable Properties of Specific Numbers
[...] 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.
[...] 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).
 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."
 T. Nagell, The diophantine equation x2+7=2n. Archiv fur Mathematik 4(13) pp. 185-187 (1960). Available from Springer
 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).
 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.
 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.
 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)
 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
 Jim Blinn, Floating-point tricks, IEEE Computer Graphics and Applications, 1997.
 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
 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)
 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.
 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.
 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.
 Gordon, Raymond G., Jr. (ed.), Ethnologue: Languages of the World (15th edition), SIL International, Dallas (2005). Online version at www.ethnologue.com
 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.
 Don N. Page, Susskind's challenge to the Hartle-Hawking no-boundary proposal and possible resolutions, 2006. arXiv:hep-th/0610199v2
 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.
 John Baez, My Favorite Numbers (web page) 2008. Includes videos and slides from three talks given in 2008 at University of Glasgow.
 My Math Forum, discussion thread, 2008 Oct 10
 N. J. A. Sloane, Eight Hateful Sequences, 2008.
 Daan van Berkel, On a curious property of 3435. (2009) arXiv:0911.3038
 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.
 WTOV, Card Users Hit With $23 Quadrillion Charge, July 15th, 2009.
 David Eberly, Fast inverse square root (revisited), 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.
 Marek Wolf, The "Skewes' number" for twin primes: counting sign changes of π2(x) − C2Li2(x), 2011. Available from arxiv.org.
 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?
 TrueNews.org, "The Origin of Life -- Evolution's Dilemma (web page), accessed 2010 April 29.
 Wolfram Alpha, "computational knowledge engine" online resource.
 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
 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:
Also, check out my large numbers
and integer sequences pages.