First page . . . Back to page 23 . . . Forward to page 25
10(2.62086...×106989) = .3-(.2-(.1-4)) = (10/3)5104
If decimal points are allowed in the digits 1 2 3 4 problem, this is the result. The subexpression .1-4 is equivalent to 104 = 10000; the subexpression .2-(.1-4) is equivalent to .2-10000 = 510000 = 5.01237274958×106989; similarly .3-x is equivalent to 3.3333...x. The idea for this was sent to me by Jim Denton (although his answer, 3.2-(.1-4), was slightly smaller).
In section 3 of [106] the authors estimate the number of "different types of universes" arising from a certain "eternal cosmic inflation" scenario, and limited by the fact that one of them is our universe, which is known to be spatially homogeneous and isotropic at large scales (a "Friedmann universe").
See also 1040, 10500, 101016, 101.877×1054, 101077, 101082, 1010166, 103.67×10281, 1010375, 105.5×10405, 101010122, and 10101.51×103883775501690.
101.55×104342944819032 = ee1013
This is the value of the "inflation factor" in a model of the inflationary universe developed by Dr. Thanu Padmanabhan, resulting from the assumption that the cosmological constant lambda equals approximately 10-8, a value arising from grand unification theories.38
103.5536897484442191...×108852142197543270606106100452735038 ≅ eee79 ≅ 10101034
The original (higher) value of the first (Riemann hypothesis true) Skewes number. It is normally written as "10101034". It was later reduced to ee27/4, which is "merely" 8.1847946207224960623437×10370. In 2005 numerical techniques were used to determine the actual value of the crossover, 1.397162914×10316. See also 1.53×101165.
101010100-1 = 999999999999...(a total of googolplex 9's)...9999
This huge number, if written out, would have a googolplex 9's in a row. Many factors are known, based on simple facts of modulo arithmetic. Since it is all 9's, it is divisible by 3. Since the number of 9's is a multiple of 2, the whole number must be a multiple of 99 (consider: 99×101=9999, 99×10101=999999, 99×1010101=99999900, and so on), and so it is divisible by 11. The same principle applies to the factors of 9999, 99999, 99999999 and any other string of 9's where the number of 9's is of the form 2a5b where a and b can each be as high as a googol (such numbers are found in sequence A003592). Additional factors of 101010100-1 can be found via Fermat's Little Theorem (see 999999). As a result there are a huge number of known factors of 101010100-1, beginning with: 3, 11, 17, 41, 73, 101, 137, 251, 257, 271, 353, 401, 449, 641, 751, ... [98].
10 to the power of googolplex. The most common name for this is googolplexian, followed by googolplexplex and googolduplex. Just as with -illion, there are many number names formed by folk etymological extension of the -plex suffix. Some are more structured than others; for example, googolduplex begins a series that continues with googoltriplex=10googolduplex, googolquadriplex, googolquinplex and so on. See also 200100 and 1010100.
In [100] (page 12), speculating on the implications of one possible resolution of a paradox brought about by the Hartle-Hawking "no-boundary" model of the universe pointed out by Susskind, the size of the universe at the end of the inflationary period would be about 101010122. The author uses the units "Mpc" (megaparsecs) although the number is so large that any length unit (such as Planck lengths, inches or furlongs) or even a volume unit (cubic parsecs, drams or bushels) could be used and the number would still be just as accurate.
See also 1040, 10500, 101016, 101.877×1054, 101077, 101082, 1010166, 103.67×10281, 1010375, 105.5×10405, 101010000000, and 10101.51×103883775501690.
10103.29994322...×10963 = eeee7.705
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 1010101000 is given.
See also 1.53×101165.
10[10(1.51×103883775501690)] = 10101010101.1
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 101.877×1054 and 105.5×10405.
1010102.0691973765×1036305 = 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.
1010101.0126×101656520 = 3 PT 1.0126×101656520 = eeeeeee
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 ee0.834032..., 143 would be eee0.471239... 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 /. The reason it never overflows is because you eventually reach a point where roundoff causes the operation X2=X×X to give X as an answer (see my uncomparably larger discussion and note the paragraph on "If A is a class 5 number").
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. There are lots of nice properties, such as a progressive gradual degradation of mantissa precision as the exponent grows (a major advantage over normal floating-point formats), reduction of all common operations to a single invertible monotonic function y=ln(x+1) with an efficient hardware implementation, and more.
This is the highest number I know of that is a limit for a computer number-representation system, apart from my own Hypercalc program which goes much higher.
Note the alternative representation "3 PT 1.0126×101656520" used here. The PT stands for "Powers of Ten", and signifies the fact that this number, expressed as a power tower, can be described as "3 powers of ten with 1.0126×101656520 at the top". Alternatively, it can be described as 4 PT 1656520.0054, which is "4 powers of ten with 1656520.0054 at the top", noting that 1656520.0054 is approximately the logarithm to base 10 of 1.0126×101656520.
1010101010000000 = 4 PT 10000000
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 10101010107.
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, 2×10④12, 2×10④20, ..., where ④ is the higher hyper4 operator. For higher levels the number is 2×10④(8n-44).
See also 101010106.8880×1014.
101010106.8880×1014 = 4 PT 6.8880×1014 = eeeee35
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 1010101010000000) and the largest I've seen apart from Graham's number.
10101010104.8293×10183230 = 5 PT 4.8293×10183230
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.
See also 6 PT 1.2826×1082.
1010101010101.2826×1082 = 6 PT 1.2826×1082
This 2 ^ ( 3 ^ ( 4 ^ ( 5 ^ ( 6 ^ ( 7 ^ ( 8 ^ 91 ) ) ) ) ) ), where each ^ is the exponent operator. It is the largest value you can get using one of each digit 1 through 9, without any symbols or punctuation. (From Louis Epstein in 2010 February84)
See also 101.0979×1019 and 5 PT 4.8293×10183230.
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.
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)
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, Wolfram MathWorld (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
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 Penguin 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 : Weisstein, Eric W. "Gamma Function." From MathWorld — A Wolfram Web Resource. 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 : see [95], page 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 : see [95], 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 : 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/archive/2004/09/rockpaperscissors.php comment by Paul Hsieh in the weblog of Michael Williams, Sep 27 2004. (Previously was at http://www.mwilliams.info/archives/004725.php)
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; also see [95] page 278.
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 : 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 : see [95], pages 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 20th.
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 21st.
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.
First page . . . Back to page 23 . . . Forward to page 25
This work is licensed under a Creative Commons Attribution 2.5
License. Details here
s.13