# Notable Properties of Specific Numbers

## Footnotes

1 : 75250000000: Glenn Elert (ed.), Mass of a Bacterium

2 : 100000000000000: Douglas F. Fix, Normal Flora

4 : 39: See [147].

5 : 10(3.4677786443...×10130): Schelter, William and the Department of Energy, Maxima (symbolic math program) There is also the SourceForge site.

6 : 5×1030: Kenneth Todar, PhD, Overview of Bacteriology

7 : 4.57936...×10917: Achim Flammenkamp, Highly Composite Numbers, web page.

8 : 25772.1300: Earth Orientation Centre, Useful Constants. (A reference formerly at http://cdsaas.u-strasbg.fr:2001/cgi-bin/resolve?AJ201486ABS is now gone; see also [170])

9 : 19: Schimmel, The Mystery of Numbers, entry for the number 19.

10 : 103.0056...×1029: Weisstein, Eric W. "Gamma Function." From MathWorld — A Wolfram Web Resource. http://mathworld.wolfram.com/GammaFunction.html

11 : 1260: Holy Bible, New International Version, Daniel 7:25 Footnote v on this verse gives the alternate translation "for a year, and two years, and half a year"; this translation is also given for Daniel 12:7.

12 : 1260: New American Bible, Revelation 12:14

13 : 1260: Amplified Bible, Revelation 12:14

14 : 60, 3600: Oxford English Dictionary [149], THIRD II. sb. 7a. shows that third has been used to mean "1/60 of a second" as far back as 1594: Euery degree .. doth containe 60 minutes, and euery minute 60 seconds, and euery second 60 thirds, &c.. A 1604 quote in the same OED entry shows "fourth" being used to mean "1/60 of a third". Another shows that three prime marks are used to denote thirds, for example: 12o 23' 34'' 45''' for "12 degrees, 23 minutes, 34 seconds, 45 thirds".

15 : 611: see [163], page 215.

16 : the Lynz: "A study of lynz..." (See also Lab6 Yearbook), website related to a group of former classmates in a British high school.

17 : 101.1×10540 and the Lynz: Weblog of James, a classmate of Adam and member of the Lab6 group.

18 : 20: see [163], page 341 (Left column, first paragraph:) "Its discovery was far from a foregone conclusion, for apart from India, Mesopotamia and the Maya civilisation, no other culture throughout history came to it by itself."

19 : 4665600000000: Wikipedia, Ritu (Indian season).

21 : 19252884016114523644357039386451: Jens Kruse Andersen, The Largest Known CPAP's (web page)

22 : 56.9612...: J. P. Benney, The number 56.96124843226, blog entry.

I had credited "Jay A. Fantini and Gilbert C. Kloepfer" for 56.9612..., but now (mid-2010) can no longer find their writing.

23 : 27: Paul Hsieh, comment on Rock-Paper-Scissors (in the weblog of Michael Williams), Sep 27 2004. (Previously was at http://www.mwilliams.info/archives/004725.php)

24 : 27: Wikipedia, Rock Paper Scissors (encyclopedia article).

25 : 27: Webpage of The World RPS Society

26 : 0.739085...: "stevo", personal communication. (MorphemeAddict -at- wmconnect com:>)

27 : 10100, [10[10(1.51×103883775501690)]|#lp3e012388]: Don Page, How to Get a Googolplex

28 : 10421: Bruce Friedman, glossary entries for the letter L at mathorigins.com

29 : 10421: From an article by J J O'Connor and E F Robertson.

30 : 0.288788...: Lee Corbin, personal communication. (lcorbin -at- uui com:>)

31 : accuracy versus precision:

If there is an amount of hay tied up in a standard-sized bale, it is both "accurate" and "precise" to call it a "bale of hay". However, if the bale is untied and the hay scattered around on the ground, it is no longer "accurate" to call it a "bale", unless you are using bale as a unit of measurement (in which case calling it "a bale of hay" would be a precise statement of the quantity of hay).

Similarly, consider the square root of 2. "1.4142135" is "precise" to 8 digits, and is also an "accurate" representation of √2, again to 8 digits. By comparison, 140/99 = 1.414141414... is a continued fraction approximation to √2. Expressing it as "1.4141414" would be "precise" to 8 digits, but is as an approximation to √2, it is only "accurate" to 4 digits.

This issue comes up a lot when observing natural phenomena and using a model that is a lot simpler than the real system. For example, the ancient Egyptians (and thence, the Romans) used 365.25 as an approximation to the tropical year, and this usage was reinforced over time by the fact that it is close to the sothic cycle as measured by the rising of Sirius as compared to the Sun; this itself is an imperfect approximation of the sidereal year. These are all slightly different numbers because in fact the Earth's axis and orbit change in subtle ways over very long periods. Thus, although the Julian calendar can be defined precisely, and the motion of the Sun against the stars can be measured precisely, both are of limited accuracy for measuring the average interval between one winter and the next.

32 : 8018018851: Neil Copeland, personal communication, Sep 2006. (neilcope -at- ihug co nz:>)

33 : 6670903752021072936960: Frazer Jarvis, Sudoku enumeration problems (web page)

34 : 2127-1 and 2521-1: The primes (2148+1)/17 and 180×(2127-1)2+1 were both found in July 1951. I have written the entries for these and a few related numbers as if it were known that the former (found by Ferrier), was discovered before the latter (found by Miller and Wheeler). In fact, it is unknown which was first. I am guessing that Ferrier was first, after considering the following: In October 1957, Miller reported that he and Wheeler found 180×(2127-1)2+1 in "early July"; On July 14, Ferrier reported that he had found (2148+1)/17; we have no evidence of a statement by Ferrier as to when he made his discovery, but it is reasonable to expect that more than two weeks passed between his discovery and his announcement.

35 : 8114118 and 535252535: Carlos Rivera, Pi such that Pi is Palprime & i = palindrome (on primepuzzles.net website)

36 : 104096: Wikipedia, Chinese numerals (encyclopedia article); also see [163] page 278.

37 : 47000000000 Wikipedia, Comoving distance (encyclopedia article).

38 : 101.55×104342944819032: T. Padmanabhan, "Inflation from quantum gravity", Phys. Letts., (1984), A104, pp 196-199.

39 : 1010166: Dave L. Renfro, Graham's Number and Rapidly Growing Functions, article in sci.math, March 4, 2002. (search for the phrase "Upper bound on the number of known universes at any specific time.")

40 : 2.255737...×1015599: Harvey Dubner, record primes with all prime digits, article in "primenumbers" Yahoo Tech Group, Feb 17 2002.

41 : http://www.numericana.com/answer/weighing.htm Gérard P. Michon, Ph.D., "The Counterfeit Penny Problem", web page (formerly at http://home.att.net/~numericana/answer/weighing.htm).

42 : Wikipedia, Change ringing (encyclopedia article).

43 : Feynman, Richard, Surely You're Joking, Mr. Feynman! (book of personal anecdotes)

45 : From the "8" lecture by John Baez[184]. Discusses the Quaternions and Octonions.

46 : Weisstein, Eric W. "Elliptic Curve Primality Proving." From MathWorld — A Wolfram Web Resource. http://mathworld.wolfram.com/EllipticCurvePrimalityProving.html

This article says that a 1 GHz processor can prove a 4769-digit prime in 3 months; thus my estimate that a 3-GHz processor can do it in 1 month.

47 : Zarko Bizaca, A reimbedding algorithm for Casson handles (section 2.4), Transactions of the American Mathematical Society, vol. 345, #2, October 1994. Also cited in Calvin Clawson, Mathematical Mysteries: The Beauty and Magic of Numbers, page 37.

49 : Wikipedia, Delta T (encyclopedia article).

50 : http://physics.nist.gov/cuu/Constants/index.html NIST reference on constants, units, and uncertainty. Gives the best known values for physical constants, including the fine-structure constant and the gravitational constant.

51 : Wikipedia, Mathematical coincidence (encyclopedia article).

52 : http://zhurnaly.com/cgi-bin/wiki/CoincidentalTaxonomy Mark Zimmermann, Coincidental Taxonomy, web page.

53 : Oxford English Dictionary [149], GREAT a. 8d: Usage of great gross goes back at least to 1640.

54 : Wikipedia, Names of large numbers (encyclopedia article).

55 : http://www.sf.airnet.ne.jp/ts/language/largenumber.html A Japanese page, titled approximately "Beyond immeasurably large numbers", which describes several systems of names for large powers of ten. Near the end is a complete table of the Avatamsaka Sutra's numbers of the form 107×2N, with Kanji names and Hiragana transliteratons.

56 : http://lass.calumet.purdue.edu/cca/jgcg/2007/fa07/jgcg-fa07-tyler.htm Eiko Tyler, Globalization and A Mathematical Journey. Lists some of the avatamsaka sutra numbers and references the Japanese source 55.

57 : Wikipedia, History of large numbers (encyclopedia article).

58 : Ian Stewart, From Here to Infinity, pp. 129-131. The same information also appeared in New Scientist magazine, issue 1941, 03 September 1994, page 18, "Fun and games in four dimensions"..

59 : http://intlpress.cn/JDG/archive/1994/39-3-491.pdf Zarko Bizaca, A Handle Decomposition of an Exotic R4, Journal of Differential Geometry, 39(3) (1994) p. 496.

61 : http://www.entsoc.org/resources/faq.htm?/print#triv1 Entomological Society of America, FAQ.

63 : Martin Gardner, The Second Scientific American Book of Puzzles & Diversions: A New Selection Simon and Schuster (1961). Coconuts: pp. 104-111.

64 : Weisstein, Eric W. "Monkey and Coconut Problem." From MathWorld — A Wolfram Web Resource. http://mathworld.wolfram.com/MonkeyandCoconutProblem.html

65 : Knuth, Donald E., Coping with finiteness. Science 194(4271) (Dec 1976), pp. 1235-1242.

67 : Wikipedia, Indian numbering system (encyclopedia article): The term crore ([Korur] in Persian) was also used in Iran until recent decades, but with the meaning of 500,000.

68 : Dale K. Hathaway and Stephen L. Brown, Fibonacci Powers and a Fascinating Triangle. The College Mathematics Journal 28(2) (Mar 1997), pp. 124-128

69 : http://oeis.org//A010048/a010048conj.png Ralf Stephan, A recurrence for the fibonomials

70 : Wikipedia, Dirac large numbers hypothesis (encyclopedia article), 2008 May 7: "Dirac noted that the ratio of the size of the visible universe [...] to the size of a quantum particle [is about] 10^{40] ..."

71 : Wikipedia, Proton (encyclopedia article).

72 : Wikipedia, HAKMEM (encyclopedia article). Describes AI Memo 239, a collection of algorithms, numerical facts and other information compiled at the MIT AI Lab in the early 1970's. Specific entries relate to the numbers 216, 239, 4.63×10170, and of course several others. A PDF file of a 1972 version of the memo is here.

73 : Wikipedia, Jargon File (encyclopedia article). Describes a glossary of slang developed by computer pioneers at MIT, Stanford and elsewhere.

74 : Wikipedia, 69105 (number) (encyclopedia article).

75 : Wikipedia, Nautical mile (encyclopedia article).

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

77 : http://www.physics.utah.edu/~cassiday/p1080/lec06.html G.L. Cassiday, The Arecibo message, course notes for Physics 1080, Univ. of Utah, 2006.

78 : http://www.johndcook.com/blog/2012/05/05/ladys-diary-1798/ John Cook, An algebra problem from 1798 (blog article), 2012 May 5.

79 : http://www.cfa.harvard.edu/iau/ECS/MPCArchive/2001/MPC_20010109.pdf Harvard-Smithsonian Center for Astrophysics minor planet center, Minor Planet Circular, page 41805 (Jan 9th 2001).

80 : Wikipedia, 5th millennium (encyclopedia article).

81 : Wilipedia, Mesoamerican Long Count calendar (encyclopedia article).

82 : Wikipedia, Monstrous moonshine (encyclopedia article).

83 : Wikipedia, Partition (number theory) (encyclopedia article).

84 : Louis Epstein, personal communication, 2005 and 2010.

85 : Weisstein, Eric W. "Prime Counting Function." From MathWorld — A Wolfram Web Resource. http://mathworld.wolfram.com/PrimeCountingFunction.html

86 : Weisstein, Eric W. "Barnes G Function." From MathWorld — A Wolfram Web Resource. http://mathworld.wolfram.com/BarnesG-Function.html

87 : Weisstein, Eric W. "K Function." From MathWorld — A Wolfram Web Resource. http://mathworld.wolfram.com/K-Function.html

88 : Weisstein, Eric W. "Panmagic Square." From MathWorld — A Wolfram Web Resource. http://mathworld.wolfram.com/PanmagicSquare.html

89 : Weisstein, Eric W. "Associative Magic Square." From MathWorld — A Wolfram Web Resource. http://mathworld.wolfram.com/AssociativeMagicSquare.html

91 : http://www.math.sjsu.edu/~hsu/courses/126/ Dr. Tim Hsu's course notes for Math 126 (San Jose State Univ., Spring 2009) had links to many important papers.

92 : Wikipedia, Rod (length), encyclopedia article.

93 : http://math.ucr.edu/home/baez/inches.html John Baez, Why are there 63360 inches per mile?, web page.

94 : http://www.alpertron.com.ar/GOOGOL.HTM Dario Alpern, Factors of 1000 numbers starting from googolplex, web page.

95 : http://www.math.uic.edu/~ronan/163 Mark Ronan, 163, the Monster and Number Theory, web page.

96 : On 4900 and the Leech lattice: The vector I describe as "(0,1,2,3,4,...,24,70)" appears to be mentioned in a 1982 paper by Conway and Sloane [145]. They discuss several vectors including a w=(1,1,1,1,1,1,1,1,1|3), another w=(0,1,2,...,23,24|70), v1=(18,39,58|17), v2=(113,312|11), v3=(118,37|9) and v4=(115,39,5|11); all of which have the property that the length of the 25-dimensional part on the left is equal to the length of the 1-dimensional part on the right. However, there are other vectors that do not have this property and I don't know what notation is being used or whether the length coincidence has any significance.

97 : http://people.virginia.edu/~mah7cd/Math552/ Mike Hill, course notes for Math 552, University of Virginia, Spring 2008.

98 : Raphie Frank, personal communication, 2010.

99 : http://wwwhomes.uni-bielefeld.de/achim/highly.txt Achim Flammenkamp, table of highly composite numbers.

100 : From the "24" lecture by John Baez[184]. The value zeta(-1)=-1/12 figures in a calculation relating to quantum ground-state energy in which an additional factor of 1/2 results in a coefficient of 1/24; Baez later says that it essentially works out to the fact that 4×6=24. I am guessing, based on my experience with the series sums use to compute the Zeta function

101 : Weisstein, Eric W. "Pi Formulas." From MathWorld — A Wolfram Web Resource. http://mathworld.wolfram.com/PiFormulas.html

102 : Martin Gardner, Mathematical Magic Show (1978) pp. 61 and 64. This is a reprint (with addenda) of a column from Scientific American (August 1967), and quoted extensively elsewhere.

105 : Wikipedia, Zero, encyclopedia article.

106 : David R. Conrad, personal communication, 2001.

107 : see [163], page 416.

109 : David Bradley (sciencebase), 40320, Such a Significant Figure, internet article.

110 : Central Council of Church Bell Ringers, First Performances and Progressive Longest Lengths (history of notable tower bell change-ringing performances).

111 : Wikipedia, xera (discussion page for the SI prefix article), 2008.

112 : New York Times, [Military Supercomputer Sets Record|http://www.nytimes.com/2008/06/09/technology/09petaflops.html], 2008 June 9th.

113 : Wikipedia, 495 (number), encyclopedia article.

115 : Here is a photo of lab6, the classroom where "The_Lynz" were actually assigned.

116 : James of Lab6, 10 years of lynz (blog article), 2008.

118 : xkcd blag, The Clarkkkkson vs. the xkcd Number, blog article with discussion, 2007.

119 : Weisstein, Eric W. "Almost Integer" From MathWorld — A Wolfram Web Resource. http://mathworld.wolfram.com/AlmostInteger.html

120 : Benjamin Vitale, "Mirror image equalities", (blog article), 2012.

121 : Charles Steevens, personal communication, 2010 Jan 14.

122 : see [163], pages 220,226.

125 : Wikipedia, Changes of the length of day (encyclopedia article).

126 : Wikipedia, Tidal acceleration (encyclopedia article).

127 : Steve Allen, Plots of deltas between time scales (web page), 2012.

129 : Wikipedia, 不可説不可説転 (hukasetsuhukasetsuten), (encyclopedia article) (in Japanese)

130 : 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.

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.