# Notable Properties of Specific Numbers

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## Bibliography

[128] 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.

[129] 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.

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

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

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

[133] 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."

[134] Claude Shannon, Prediction and Entropy of Printed English, Bell System Technical Journal 28 pp. 50-64 (1951).

[135]
T. Nagell, The diophantine equation x^{2}+7=2^{n}.
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] Tibor Rado, On non-computable functions, Bell System Technical Journal 41(3) pp. 877-884, 1962.

[138] 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.

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

[140] 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).

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

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

[143] 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.

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

[145] 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

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

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

[148] 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.

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

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

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

[152] 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)

[153] 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.

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

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

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

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

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

[159]
Maurice Mignotte and Attila Pethö,
On the system of diophantine equations
x^{2}-6y^{2}=-5 and x=2z^{2}-1. Mathematica Scandinavica
76, pp. 50-60 (1995). Available from the publisher
here

[160] 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

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

[162] David Singmaster, Chronology of Recreational Mathematics, 1996.

[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
x^{2}-6y^{2}=-5, x=2z^{2}-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=mc^{2} (handout, PDF
file), 2002.

[177] Eric Balandraud, Calculating the Permutations of 4D Magic Cubes, 2003.

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

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

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

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

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

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

[184] Sean M. Carroll and Jennifer Chen, Spontaneous Inflation and the Origin of the Arrow of Time, PDF on arXiv

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

[186]
Maohua Le, On the diophantine system
x^{2}-Dy^{2}=1-D AND x=2z^{2}-1. Mathematica Scandinavica
95, pp. 171-180 (2004). Available from the publisher
here.

[187] 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

[188]
Gordon, Raymond G., Jr. (ed.), Ethnologue:
Languages of the World (15^{th} edition), SIL International, Dallas
(2005). Online version at
www.ethnologue.com

[189] 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.

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

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

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

[193] 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.

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

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

[196]
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.

[197] David de Neufville, personal correspondence.

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

[199] 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.

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

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

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

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

[204]
CNN Beat 360, Anderson Cooper Daily Podcast for July
15^{th}, 2009.

[205]
Huffington Post,
Man Charged 23 Quadrillion...,
July 15^{th}, 2009.

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

[207]
WMUR TV-9 (Manchester NH),
Man's Debit Card Charged $23 Quadrillion...,
July 15^{th}, 2009.

[208] http://www.astro.ucla.edu/~wright/cosmology_faq.html Edward L. Wright, Frequently Asked Questions in Cosmology (web page), 2009.

[209]
WTOV,
Card Users Hit With $23 Quadrillion Charge,
July 15^{th}, 2009.

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

[211] Jeffrey Hankins, personal correspondence, 2010.

[212] 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.")

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

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

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

[216] Ivan Panchenko, personal correspondence, 2011.

[217]
Marek Wolf,
The "Skewes' number" for twin primes:
counting sign changes of π_{2}(x) − C_{2}Li_{2}(x),
2011. Available from arxiv.org.

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

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

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

[221] 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).

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

[223] "Pat's Blog", Before there were four fours..., 2012.

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

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

[226] 27: This is close.

[227] 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.

[228] 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

[229] "MikeMcl", decimal.js, JavaScript library for handling large numbers.

[230] 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

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

[232] Aarex Tiaokhiao, magna_numerus.js, JavaScript library for handling large numbers (also includes confractus_numerus.js and logarithmica_numerus_lite.js)

[233] "Patashu" (Timothy Stiles), break_infinity.js, JavaScript library for handling large numbers (checkout a commit prior to 2019 March to get the deprecated break_break_infinity.js).

[234] "Patashu" (Timothy Stiles), break_eternity.js, JavaScript library for handling large numbers.

[235] James Read and Baptiste Le Bihan. "The landscape and the multiverse: What's the problem?". Synthese. 199(3–4) 7749–7771 (2021)

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
5×10^{11}
10^{18}
5.4×10^{27}
10^{40}
5.21...×10^{78}
1.29...×10^{865}
10^{40000}
10^{9152051}
10^{1036}
10^{1010100}
— —
footnotes
Also, check out my large numbers
and integer sequences pages.

s.27