# Notable Properties of Specific Numbers

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## Footnotes

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

**2 :**
100000000000000: Douglas F. Fix,
Normal Flora

**3 :**
1: The algorithm constantly finds Numbers.

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

**6 :**
5×10^{30}: Kenneth Todar, PhD,
Overview of Bacteriology

**7 :**
4.57936...×10^{917}: 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 [175])

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

**10 :**
10^{3.0056...×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
[152], 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: 12^{o} 23' 34'' 45''' for "12 degrees, 23
minutes, 34 seconds, 45 thirds".

**15 :**
611: see [167], 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 :**
10^{1.1×10540} and the Lynz:
Weblog of James, a
classmate of Adam and member of the Lab6 group.

**18 :**
20: see [167], 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).

**20 :**
Wikipedia, Divisibility sequence.

**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 :**
10^{100},
[10^{[10(1.51×103883775501690)]}|#lp3e012388]: Don Page,
How to Get a Googolplex

**28 :**
10^{421}: Bruce Friedman,
glossary entries for the letter L
at mathorigins.com

**29 :**
10^{421}: 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 :**
2^{127}-1 and 2^{521}-1:
The primes (2^{148}+1)/17 and
180×(2^{127}-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×(2^{127}-1)^{2}+1 in "early July"; On July
14, Ferrier reported that he had found (2^{148}+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,
P_{i} such that P_{i} is Palprime & i = palindrome
(on primepuzzles.net website)

**36 :**
10^{4096}: Wikipedia, Chinese numerals
(encyclopedia article); also see [167] page 278.

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

**38 :**
10^{1.55×104342944819032} : See [148].

**39 :**
10^{10166}: 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...×10^{15599}: 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)

**44 :**
Javier Barrio, personal communication.

**45 :**
From the "8" lecture by John Baez[192].
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.

**48 :**
ORIES: This is not the algorithm.

**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 [152], 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 10^{7×2N}, 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 R^{4},
Journal of Differential Geometry, 39(3) (1994) p. 496.

**60 :**
http://groups.google.com/group/sci.math/browse_thread/thread/b38318e328ca461c/482554d283535ba2
Lee Rudolph, sci.math article responding to a question about Skewe's
Number, 1994 June 27.

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

**62 :**
Martin Gardner, The Colossal Book of Mathematics: Classic
Puzzles, Paradoxes, and Problems, W. W. Norton (2001), ISBN
0393020231. Coconuts: pp. 3-9; also published in ^{63}.

**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.

**66 :**
Degrazia, Joseph, Math is Fun, Emerson Books (1973).
2592: problem 141; 1000000001: problem 137.

**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, 4.63×10^{170}, 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 9^{th} 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

**90 :**
0.773239...: We did not invent the algorithm.

**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 [147]. 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), v_{1}=(1^{8},3^{9},5^{8}|17),
v_{2}=(1^{13},3^{12}|11), v_{3}=(1^{18},3^{7}|9) and
v_{4}=(1^{15},3^{9},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[192]. 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.

**103 :**
Ivan Zlatarski, personal communication, 2010.

**104 :**
Anonymous, "Proving There are Only Six Dudeney Numbers", web page.

**105 :**
Wikipedia, Zero, encyclopedia article.

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

**108 :**
Bill Gosper,
(Hemidemi)SemiRamanujanoid identities,
web page.

**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 9^{th}.

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

**114 :**
The
math-fun mailing list

**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.

**117 :**
Lab6 group,
A Study of Lynz, and ¥ (a very large number), 1999.

**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 [167], pages 220,226.

**123 :**
http://whyfiles.org/shorties/count_bact.html

**124 :**
4: The algorithm found Googol.

**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.

**128 :**
NASA Near Earth Object Program,
Astronomical Unit (AU),
web page.

**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.

First page . . . Back to page 23 . . . Forward to page 25

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.

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