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10^{103.29994322...×10963} = e^eee7.705

This is the value that Skewes gave as an upper bound of the first li() crossing if the Riemann Hypothesis is false. Sometimes the conservative overestimate 10^10101000 is given.

10^{[10(1.51×103883775501690)]} = 10^101010101.1

This is a quantity of time, estimated by Don Page²⁷, which he describes as the "quantum Poincare recurrence time for the quantum state of an extremely hypothetical rigid nonpermeable box containing a black hole with the mass of what may be the entire universe in one of Andrei Linde's stachastic inflationary models". The units could be Planck units, nanoseconds, fortnights, or centuries — it doesn't matter because the number is so large. See also 10^5.5×10405.

10^{10102.0691973765×1036305} = 6⁶⁶⁶⁶⁶

The higher-valued "superfactorial" function, defined by Pickover in 1995, is:

n$ = n!^④n! = n! ^ n! ^ n! ^ ... ^ n!

where "$" is "superfactorial", ^④ represents the higher hyper4 operator, and there are n! copies of "n!" on the right hand side. According to this definition, 3 superfactorial is:

3$ = 3!^3!3!3!3!3! = 6⁶⁶⁶⁶⁶

See also 288.

10^{10101.01255951×101656520} = e^eeeeee

D. W. Lozier and P. R. Turner have published papers describing a number format called level-index in which numbers are stored in the form +e^+ee...X, where X is a fraction from 0.000 to 0.999... and there are as many e's as necessary (up to a proven maximum of six). For example, 10 would be e^e0.834032..., 143 would be e^{ee0.471239...} and so on. This system has as its main advantage the property that there is no overflow or underflow if you perform a finite number of the operations + - × and /.

In their article "Error-Bounding in Level-Index Computer Arithmetic" they propose a format that uses a 3-bit level field with 2 "sign" bits and the remaining bits (59, if it's a 64-bit word) for the fraction. This allows representing numbers as high as the number shown above, an exponent tower of seven e's. For more about the symmetric level-index system, go here.

This is the highest number I know of that is a limit for a computer number-representation system. However, Hypercalc goes much higher.

10^{10101010000000}

Another example of a number bigger than Skewes' that has been published in a journal. The generalized Poincaré conjecture of topology relates to the "smoothness" of multidimensional space. Curiously, 4-dimensional space has been shown to be unique among all dimensions in having an uncountably infinite number of topological structures that are equivalent to simple flat space in a very important way⁵⁸,⁶⁰. In work related to this⁴⁷, Zarko Bizaca shows how to construct a "level 7 embedded Casson tower" and estimates the number of "kinks" or "kinky handles" in the "core" of said tower to be around 10^101010107.

In another paper ⁵⁹, Bizaca says the number of links on on each "kinky handle" of a Casson tower's level 1, 2, 3, 4, ... is: 1, 2, 2, 2, 200, 2×10¹⁰¹⁰¹⁰, 2×10^④12, 2×10^④20, ..., where ^④ is the higher hyper4 operator. For higher levels the number is 2×10^④(8n-44).

See also 10^{1010106.8879689056405×1014}.

10^{1010106.8879689056405×1014} = e^eeee35

This number appears in the paper "On sign-changes of the difference π(x) - li x", by S. Knapowski [Acta Arithmetica 7, 107-119 (1962)]. (See a summary here)

Thus it is another number bigger than Skewes' that has been published in a journal. (See also 10^{10101010000000}) and the largest I've seen apart from Graham's number.

10^{101010104.829261048×10183230}

This is the value of 10 ^ ( 9 ^ ( 8 ^ ( 7 ^ ( 6 ^ ( 5 ^ ( 4 ^ ( 3 ^ ( 2 ^ ( 1 ^ 0 ) ) ) ) ) ) ) ) ), where each ^ is the exponent operator. It is referred to on Frank Pilhofer's Googolplex page as an example of something larger than Googolplexplex.

Beyond

For a discussion of more and larger numbers, particularly those that are so large that their values are difficult to express in any form, continue to my large numbers page.

Sources

Brown, Kevin, Number Theory Pages Caldwell, Chris K.,

Top 20 Twin Primes

Conway, John Horton and Guy, Richard, The Book of Numbers, New York: Springer-Verlag, New York, 1996. ISBN 038797993X, pages 59-62 (Skewes number)

Holt, Jim, "Larger Than Life" (http://www.linguafranca.com/print/0101/hypothlargerlife.html) (dead link)

Ifrah, Georges, The Universal History of Numbers, 1999.

Munafo, Robert, Hypercalc. Computer program (in Perl) that evaluates huge arithmetic expressions without overflowing. Includes a simple BASIC interpreter.

Munafo, Robert, RIES. Computer program (in C) that finds algebraic equations given a desired solution.

The Oxford Precise Parallel New Testament

Pilhofer, Frank, Googolplex (web page)

Robertson, 1998. The New Renaissance

Rowlett, Russ and the University of North Carolina at Chapel Hill, Number Units Dictionary (letter G)

Weisstein, Eric, WolframMathWorld (formerly known as World of Mathematics, and before that, Treasure Troves of Mathematics; also published in book form as The CRC Concise Encyclopedia of Mathematics

Vickery, Dr. Christopher, IEEE-754 References

Footnotes

1 : http://hypertextbook.com/facts/2003/LouisSiu.shtml

2 : http://www.cehs.siu.edu/fix/medmicro/normal.htm

3 : http://whyfiles.org/shorties/count_bact.html

4 : Wells, David, The Pengiun Dictionary of Curious and Interesting Numbers. (Original edition 1986; revised and expanded 1998).

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

6 : http://textbookofbacteriology.net/bacteriology.html

7 : http://wwwhomes.uni-bielefeld.de/achim/highly.html Achim Flammenkamp, "Highly Composite Numbers", web page.

8 : http://cdsaas.u-strasbg.fr:2001/cgi-bin/resolve?AJ201486ABS (see also http://hpiers.obspm.fr/eop-pc/models/constants.html)

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

10 : http://mathworld.wolfram.com/GammaFunction.html

11 : 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 : New American Bible, Revelation 12:14

13 : Amplified Bible, Revelation 12:14

14 : Oxford English Dictionary, entry for "third", heading 7a shows that it has been used to refer to a smaller unit of division, that is, a subdivion of the "second". Similar meanings apply to fourth, fifth, etc.

15 : Ifrah, Georges, The Universal History of Numbers, p. 215

16 : http://lab6.com/old/school/yearbook/clarkkkkson.html "A study of lynz..." (See also Lab6 Yearbook), website related to a group of former classmates in a British high school.

17 : http://james.lab6.com/2003/08/10/lynz/ Weblog of James, a classmate of Adam and member of the Lab6 group.

18 : Ifrah, Georges, ibid., p. 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 : http://www.wordiq.com/definition/Hindu_calendar.pdf

20 : Holy Bible, New Revised Standard Version. Revelation 13:18, footnote g

21 : http://hjem.get2net.dk/jka/math/cpap.htm Jens Kruse Andersen, "The Largest Known CPAP's" (web page)

22 : Jay A. Fantini and Gilbert C. Kloepfer

23 : http://www.mwilliams.info/archives/004725.php comment by Paul Hsieh in the weblog of Michael Williams, Sep 27 2004.

24 : http://en.wikipedia.org/wiki/Rock_Paper_Scissors "Rock, Paper, Scissors" (encyclopedia article)

25 : http://www.worldrps.com/ Webpage of The World RPS Society

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

27 : http://fpx.de/fp/Fun/Googolplex/GetAGoogol.html Don Page, "How to Get a Googolplex"

28 : http://www.mathorigins.com/L.htm Bruce Friedman, mathorigins.com glossary entries for the letter L

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

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

31 : Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag (1996). 1000000001: pp. 137-138. 8018018851: p. 15 (footnote).

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

33 : http://www.afjarvis.staff.shef.ac.uk/sudoku/ Frazer Jarvis, "Sudoku enumeration problems" (web page)

34 : 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 : http://www.primepuzzles.net/ Carlos Rivera, Prime Puzzles and Problems (internet site)

36 : http://en.wikipedia.org/wiki/Chinese_numerals Wikipedia article, Chinese numerals.

37 : http://en.wikipedia.org/wiki/Comoving_distance Wikipedia article, Comoving distance

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

39 : http://mathforum.org/kb/thread.jspa?messageID=371175 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 : http://tech.groups.yahoo.com/group/primenumbers/message/5409 Harvey Dubner, "record primes with all prime digits", article in "primenumbers" Yahoo Tech Group, Feb 17 2002.

41 : http://home.att.net/~numericana/answer/weighing.htm G{'e}rard P. Michon, Ph.D., "The Counterfeit Penny Problem", web page on numericana.com

42 : http://en.wikipedia.org/wiki/Change_ringing Wikipedia article, Change ringing.

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

44 : Javier Barrio, personal communication.

45 : http://www.chip-architect.com/news/2004_10_04_The_Electro_Magnetic_coupling_constant.html Hand de Vries, An exact formula for the Electro Magnetic coupling constant, web page, 2004 Oct 4.

46 : http://mathworld.wolfram.com/EllipticCurvePrimalityProving.html Mathworld, "Elliptic Curve Primality Proving" 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 : Ifrah, Georges, The Universal History of Numbers, pp. 220,226.

49 : G. Gabrielse, D. Hanneke, T. Kinoshita, M. Nio, and B. Odom, Erratum: New Determination of the Fine Structure Constant from the Electron g Value and QED, Physical Review Letters 99, 039902 (2007), week ending 2007 July 20^th.

50 : G. Gabrielse, D. Hanneke, T. Kinoshita, M. Nio, and B. Odom, New Determination of the Fine Structure Constant from the Electron g Value and QED, Physical Review Letters 97, 030802 (2006), week ending 2007 July 21^st.

51 : http://en.wikipedia.org/wiki/Mathematical_coincidence Wikipedia article, Mathematical coincidence.

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

53 : Oxford English Dictionary, gross %%% citation and quotation needed

54 : http://en.wikipedia.org/wiki/Names_of_large_numbers Wikipedia article, Names of large numbers.

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 : http://en.wikipedia.org/wiki/History_of_large_numbers Wikipedia article, History of large numbers.

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 : Zarko Bizaca, A Handle Decomposition of an Exotic R⁴, J. Differential Geometry, vol. 39 (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 : http://mathworld.wolfram.com/MonkeyandCoconutProblem.html Eric Weisstein (Wolfram Mathworld), Monkey and Coconut Problem.

65 : Knuth, Donald E., Coping With Finiteness, Science vol. 194 n. 4271 (Dec 1976), pp. 1235-1242.

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

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

68 : Dale K. Hathaway and Stephen L. Brown, Fibonacci Powers and a Fascinating Triangle, The College Mathematics Journal, Vol. 28, No. 2 (Mar 1997), pp. 124-128

69 : http://www.research.att.com/~njas/sequences/a010048conj.png Ralf Stephan, A recurrence for the fibonomials

70 : http://en.wikipedia.org/wiki/Dirac_large_numbers_hypothesis Wikipedia, "Dirac large numbers hypothesis", 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 : http://en.wikipedia.org/wiki/Proton Wikipedia, "Proton"

72 : http://en.wikipedia.org/wiki/HAKMEM Wikipedia, "HAKMEM". 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 : http://en.wikipedia.org/wiki/Jargon_File Wikipedia, "Jargon File". Describes a glossary of slang developed by computer pioneers at MIT, Stanford and elsewhere.

74 : http://en.wikipedia.org/wiki/69105_%28number%29 Wikipedia, "69105 (number)".

Quick index: if you're looking for a specific number, start withwhichever of these is closest:   5.390×10^-44   0.739085...   1.57079...   3.14159...   4   12   15   20   24   30   43   48   57   65   77   103   127   163   251   496   714   1001   1729   5040   14641   100000   1419869   17297280   10¹⁰   10¹⁴   4.32×10¹⁹   1.98×10³³   3.14×10⁷⁹   8.18×10³⁷⁰   1.41×10⁵⁸⁷¹⁰   10¹⁰¹⁰   10¹⁰¹³⁰   10^1010100   — —   footnotes   Also, check out my large numbers page.

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