Notable Properties of Specific Numbers
First page . . . Back to page 21 . . . Forward to page 23 . . . Last page (page 25)
10^{8.0723047260281×10153} = 4^{444}
The highest value achievable in the "four 4's" puzzle. (This is the puzzle that asks you what numbers you can make using four 4's and the common operations on a calculator, for example, 1=^{44}/_{44}, 2=^{4}/_{4}+^{4}/_{4}, 3=^{4+4+4}/_{4}, 4=^{4}√4/4)×4, etc.) See also 10^{1.0979×1019}.
A somewhat lower estimate of the number of possible universe histories given by Dave L. Renfro and calculated by a different method (he estimates the Planckunit volume of the universe at 10^{123}, the number of particles at 10^{80} and the universe age at 10^{41} times an "interval" of 10^{24} seconds).^{39} See also 10^{1.877×1054}, 10^{3.79×10281} and 10^{5.7×10405}.
(my "simple alternate universe count")
A highlysimplified formula to compute the number of possible universes. N = e^{v n} where N is the number of possible universes, n is the number of fundamental particles in the universe and v is the number of particles that could be fit in the universe if it were packed full of particles. See also 10^{1.877×1054}.
This is e^{e661}, an improved (but erroneous) upperbound for the π(x) vs. li(x) problem (the higher "Skewes' number"), by Alan Turing in an unpublished manuscript. It was corrected to 10^{2.6654...×10536} by Cohen and Mayhew in 1965.
An estimate of the number of distinct universes in the "string theory landscape", given by [201] in section 5, and assuming that the "maximal number of observable efolds" is about 290. ^{121}
See also 10^{40}, 10^{500}, 10^{1016}, 10^{1.877×1054}, 10^{1077}, 10^{1082}, 10^{10166}, 10^{3.79×10281}, 10^{5.7×10405}, 10^{1010000000}, 10^{1010122}, and 10^{101.51×103883775501690}.
(my "alternate universe count")
The factorial of the singleperturbation count, a highly theoretical estimate of the number of different ways all the particles in the known universe could be randomly shuffled at each moment in time since the universe's creation. In quantum mechanics, it is comparable to the number of universe timeline wavefunctions that exist simultaneously from the viewpoint of an observer outside our universe. Another way of saying the same thing is that, if the universe is being created over and over again, it would take (on average) this many repetitions before one would expect to get an exact recurrence of "our" universe.
This estimate is entirely arbitrary and omits many details (relativistic curvature, dark matter and dark energy, Pauli exclusion, etc.) because there is probably an even greater discrepancy between the "known" and "actual" size of the universe.
See also 10^{1.877×1054}, 10^{10166}, 10^{3.79×10281} and 10^{101.51×103883775501690}.
This is e^{e1236}, an improvement on the higher "Skewes' number" (and a correction of Alan Turing's 10^{5.0867...×10286}) published by Cohen and Mayhew in 1965.
The number of Adam Clarkson's lynz as of the 10^{th} of August, 2003 (and due the following day) as referenced on a classmate's blog^{17}.
This is the factorial of 10^{666}, and is called the leviathan number by Clifford Pickover [185]. The word leviathan refers to a whale or seamonster. Biblical references to "leviathan" are all in the Old Testament, although 666 is more commonly associated with the last book of the New Testament.
10^{(1.889876...×101982)} = 2^{26579×203}×10^{26580}
(the everincreasing number of Adam Clarkson's "Lynz")
The number of lines of text one Adam Clarkson will owe his (former) high school chemistry teacher on 22^{nd} September 2016. The story runs as follows^{16}:
In February (on the 26^{th}, to be precise, since it's a good day to celebrate, num' sein?) of 1998, a chemistry teacher gave a set of lines to one of his students, Adam Clarkson. The lines read, "I must always tuck my shirt in whilst participating in a Chemistry Lesson".^{115} He had to do a hundred of them. However, the clever part was that if he didn't do them by the next day, they would double, and if they weren't done by the day after that, they would double again. We pointed out that the lines would become too great to do pretty soon, but this didn't stop the teacher giving them. . . .
Daily doubling is a very effective way to outpace anyone's ability at nearly any task; see the chess legend for another example. After one week the assignment had grown to 12,800 lines, which doesn't seem too bad, relatively speaking, but another week makes it 1,638,400. By the 26^{th} March (one month after the initial assignment) it would have been 26,843,545,600 (more lines than in all the books of the school's library), and by 26^{th} April Adam would have had to enlist the assistance of the entire world's population, writing over 8 billion lines apiece to complete the total 100×2^{59} ≈ 5.7646×10^{19} lynz. This is slightly more than the number of combinations of a Rubik's cube, coincidentally the subject of another classic case of human innumeracy.
This alone would be a quintessential legend of the human struggle to understand large numbers, but the story doesn't end there^{16}:
. . . [On the 17^{th}] September 1998, 7 months late, due to a confused and old man apparently mishearing us (we tried to inform him of the vastness of the Lynz, but he didn't want to hear, and so he said "If they're not on my desk by tomorrow, they'll square!"), the lines were squared every day they weren't done. To this day, they still haven't been done.
Since the 17^{th} September was 203 days after the 26^{th} of February, the lines had been doubled 203 times — on that day the assignment was 100×2^{203} ≈ 1.286×10^{63} lines, a little over one vigintillion, and due the next day. This was more or less equal to the capacity of the known universe to produce handwritten lines, but only if all the galaxies were converted into paper and ink. There was no chance of doing these Lynz by the next day, and so the squaring commenced.
On the 18^{th} and each day thereafter, the number was squared, so the assignment on the 18^{th} September was (100×2^{203})^{2} = 100^{2}×2^{2×203}, about 1.65×10^{126}: in one day they went from a vigintillion to being much larger than a googol. The day after that it was 100^{4}×2^{4×203}, or about 2.73×10^{252}, and so forth. Within days they blew past the short and longscale centillion and millillion; within a week or two more they had grown beyond the odds against Ford and Arthur's rescue, the Hamlet monkey number, a millimillillion, and the various recordsize primes. These numbers are many millions of digits long, meaning that it would have taken Adam millions of sheets of paper just to write down how many lines were due.
Because they are squared daily, the number of digits of the number of lynz doubles each day. By their first anniversary the Lynz were about 10^{3.689×1050}, just over G. H. Hardy's estimate of the number of possible chess games; by 8^{th} August 1999 they surpassed a googolplex. By their second anniversary (26^{th} February 2000), they had become so large that if you tried to write not the lines themselves, but merely the number of lines as a normal decimal number (that is, without using scientific notation) you'd be writing a number over 10^{160} digits long, a feat which could not be accomplished even if you could fit a vigintillion digits on each particle in the observable universe. By 2003 the length of the assignment had grown^{17} to the point where it was just "somewhat larger" than my rough estimate of the number of ways the universe's history could be shuffled. By their tenth anniversary^{116} the Lynz were in excess of 10^{101000}, and the "Clarkkkkson" function based on them^{117} was generally regarded^{118} to be fastergrowing than most anything out there. As the author points out,
[...] with the ticking timebomb that is [The Lynz] squaring every day, it's got a Gmailstyle claim to infinity.
To make clear the calculation "2^{26579×203}×10^{26580}", this table shows the Lynz daybyday, then by months, then by years:

So, on the 22^{nd} September 2016 the number of lines due will be 2^{26579×203}×10^{26580}, which is about 10^{(1.889876×101982)}.
This reallife story invokes a similar respect for the innumeracy of common people to that described in the ancient chess legend; see also 2.315×10^{16}, 43252003274489856000 and 10^{137}.
10^{(2.62086...×106989)} = .3^{(.2(.14))} = (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 10^{4} = 10000; the subexpression .2^{(.14)} is equivalent to .2^{10000} = 5^{10000} = 5.01237274958×10^{6989}; similarly .3^{x} is equivalent to 3.3333...^{x}. The idea for this was sent to me by Jim Denton (although his suggestion, 3^{.2(.14)}, was slightly smaller).
In section 3 of their paper "How many universes are in the multiverse?" [201] Linde and Vanchurin 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 10^{40}, 10^{500}, 10^{1016}, 10^{1.877×1054}, 10^{1077}, 10^{1082}, 10^{10166}, 10^{3.79×10281}, 10^{10375}, 10^{5.7×10405}, 10^{1010122}, and 10^{101.51×103883775501690}.
This is the approximate value of 9^{999}. Using Mathematica version 9, Robert G. Wilson was able to compute the first 100 digits and the last 100 digits of this huge number; see OEIS sequence A243913. See also 4.28...×10^{369693099}.
In an answer on Philosophy stackexchange [216], I describe a hypothetical situation in which a person (who cannot age or want from any hardships) is watching a display that shows a number of the form 10^{N} in normal notation (that is, a 1 followed by N zeros). As time goes on, N gets steadily larger (the number of zeros keeps increasing). The question is, "how big could the number get and the watcher still be able to directly perceive each displayed power of 10 in a distinct way?". I believe there is a limit to the mental capacity of this hypothetical observer. If that mental capacity is equivalent to a billion letters written on paper, then the person could "count the zeros" until there are 26^{1000000000} zeros, and so the number being displayed would be 10^{261000000000}. Rewritten in the form of 10^{10x}, this comes out to about 10^{9.35...×101414973347}.
See also 10^{1016} and 10^{1018}.
10^{3×103000000000+3} ≈ 10^{10109.477121...}
Jonathan Bowers' gigillion, defined precisely as 10^{3×103×109+3}. This is part of a sequence, preceded by megillion = 10^{3×103×106+3} and followed by terillion = 10^{3×103×1012+3}. See also 97pt10^{10000000000}.
10^{101010} = 10^{1010000000000} = 10^{④}4
A powertower of four 10's, written "10^{④}4" using my hyper4 notation or "10↑↑4" using Knuth's uparrow notation. See also 97pt10^{10000000000}.
10^{1.55×104342944819032} = e^{e1013}
This is the value of the "inflation factor" in a model of the inflationary universe developed by Dr. Thanu Padmanabhan [149], resulting from the assumption that the cosmological constant lambda equals approximately 10^{8}, a value arising from grand unification theories.
10^{3.5536897484442191...×108852142197543270606106100452735038} ≈ e^{ee79} ≈ 10^{101034}
(Skewes' number)
The Skewes' number, in the general sense refers to the lowest value x for which the primecounting function π(x) is larger than the logarithmic integral function li(x).
e^{ee79} is the original (higher) value of the first (Riemann hypothesis true) Skewes' number, published in a 1933 paper. It is normally written as "10^{101034}". It was later reduced to e^{e27/4}, which is "merely" 8.1847946207224960623437×10^{370}. In 2005 numerical techniques were used to determine the actual value of the crossover, 1.397162914×10^{316}. See also 1.53×10^{1165}, 10^{103.3×10963} and 4pt6.8880×10^{14}.
10^{1010100}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, for the same reason that 10^{10}1 = 9999999999 = 99×101010101; and because it is divisible by 99, it is divisible by 11. The same principle lets us add factors of 9999 = 10^{4}1, 99999 = 10^{5}1, 99999999=10^{8}1, and any number of the form 10^{2a5b}1 where a and b can each be as high as a googol (such numbers are found in OEIS sequence A3592). Additional factors of 10^{1010100}1 can be found via Fermat's Little Theorem (see 999999). As a result there are a huge number of known factors of 10^{1010100}1, beginning with: 3, 11, 17, 41, 73, 101, 137, 251, 257, 271, 353, 401, 449, 641, 751, ... [181].
("googolplexplex")
10 to the power of googolplex.
Just as with illion, there are many number names formed by folk etymological extension of the plex suffix, such as millionplex for 10^{1000000}.
A common name for 10^{googolplex} is googolplexian (seen in Internet searches) but I suspect googolplexplex is more commonly used by folks who try to come up with their own name. I have also seen googolduplex proposed by folks who want to extend the names further. googolduplex begins a series that continues with googoltriplex=10^{googolduplex}, googolquadriplex, googolquinplex and so on.
See also 200^{100} and 10^{10100}.
In [156] (page 8), Don Page estimates the Poincare recurrence time of a black hole of a mass equivalent to the visible universe to be "10^{1010102.08} Planck times, millennia, or whatever." It can be understood as a period of time equal to the number of possible "macrostates" of the system inside the black hole, which is e^{N} where N is the number of "microstates", and this in turn is equal to e^{es} where s is the maximum possible entropy of the system. s is approximately (r/l_{p})^{2} where r is the radius of the universe and l_{p} is the Planck length. Using a current (2014) value for the size of the visible universe we would get (4.45×10^{26}/(1.62×10^{35}))^{2} or about 7.5×10^{122} in the exponent. This number is also described in Numberphile's video The LONGEST Time. See also 10^{101010101.1}.
In [189] (page 12), speculating on the implications of one possible resolution of a paradox brought about by the HartleHawking "noboundary" model of the universe pointed out by Susskind, the size of the universe at the end of the inflationary period would be about 10^{1010122}. 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 10^{40}, 10^{500}, 10^{1016}, 10^{1.877×1054}, 10^{1077}, 10^{1082}, 10^{10166}, 10^{3.79×10281}, 10^{10375}, 10^{5.7×10405}, 10^{1010000000}, and 10^{101.51×103883775501690}.
10^{103.29994322...×10963} = e^{eee7.705}
(the Higher "Skewes' number")
In 1955, Skewes gave this as an upper bound of the first π(x)  li(x) crossing if the Riemann Hypothesis is false (see the first Skewes number entry for a fuller description). Sometimes the conservative overestimate 10^{10101000} is given. It was improved to 10^{2.6654...×10536} by Cohen and Mayhew in 1965, before actual computational results (starting with Lehman's 1.53×10^{1165}) took over.
See also 4pt6.8880×10^{14}.
10^{[10(1.51×103883775501690)]} = 10^{101010101.1}
(Don Page's alternate universe count)
This is a quantity of time, estimated by Don Page in [156] (page 8) and ^{27}, 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. Numberphile did a video on it: The LONGEST Time. See also 10^{1.877×1054}, 10^{5.7×10405}, and 10^{101.7×10120}.
10^{10106.8880×1014} = 3pt6.8880×10^{14} = e^{eee35}
Wolf's incorrect value [211] of the Knapowski number.
10^{10102.069197...×1036305} = 3pt2.069...×10^{36305} = 6^{66666}
(Pickover's superfactorial of 3)
The highervalued "superfactorial" function, defined by Pickover in 1995, is:
n$ = n!^{④}n! = 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!} = 6^{66666}
See also 288 and 22pt1.84×10^{33}.
10^{10101.0126×101656520} = 3pt1.0126×10^{1656520} = e^{eeeeee}
(range of levelindex representaton)
D. W. Lozier and P. R. Turner have published papers describing a number format called levelindex 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 /. The reason it never overflows is that for sufficiently high X, roundoff causes the operation X^{2}=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". The power tower paradox discussion is also relevant.).
In their article "ErrorBounding in LevelIndex Computer Arithmetic" they propose a format that uses a 3bit level field with 2 "sign" bits and the remaining bits (59, if it's a 64bit word) for the fraction. This allows representing numbers as high as the number shown above, a "powertower" of seven e's. For more about the symmetric levelindex 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 floatingpoint 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 numberrepresentation system, apart from my own Hypercalc program which goes much higher.
Note the alternative representation "3pt1.0126×10^{1656520}" 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 1.0126×10^{1656520} at the top". Alternatively, it can be described as 4pt1656520.0054, which is "4 powers of ten with 1656520.0054 at the top", noting that 1656520.0054 is approximately the logarithm to base 10 of 1.0126×10^{1656520}.
10^{10101010000000} = 4pt10000000
Another example of a number bigger than Skewes' that has been published in a journal (in 1994). The generalized Poincaré conjecture of topology relates to the "smoothness" of multidimensional space. Curiously, 4dimensional 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 way^{58},^{60}. In work related to this^{47}, 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 each "kinky handle" of a Casson tower's level 1, 2, 3, 4, ... is: 1, 2, 2, 2, 200, 2×10^{101010}, 2×10^{④}12, 2×10^{④}20, ..., where ^{④} is the higher hyper4 operator. For higher levels the number is 2×10^{④}(8n44).
See also 10^{1010106.8880×1014}.
10^{1010106.8880×1014} = 4pt6.8880×10^{14} = e^{eeee35}
The Knapowski number
This number appears in the paper "On signchanges of the difference π(x)  li x", by S. Knapowski [137]. Knapowski proves that for all x larger than this value, the number of crossings of π(x) and li x (see Skewes') is greater than ln(ln(ln(ln(x))))/e^{35}.
Thus it is a number bigger than both of Skewes' numbers that has been published in a journal, and it is the largest I've seen apart from the various versions of Graham's number. Since Skewes' numbers were published in 1933 and 1955 and the GrahamRothschild_number was published in 1971, there was a period during which Knapowski's number "held the record" for largest number explicitly mantioned in a published academic paper. (See also 10^{10101010000000}, which is from 1994 and never held the record).
The Knapowski paper is mentioned briefly by Wolf [211] who erroneously says that it is e^{eee35}, a powertower with one fewer e.
10^{101010104.8293×10183230} = 5pt4.8293×10^{183230}
This is 10^(9^(8^(7^(6^(5^(4^(3^(2^(1^0))))))))), where each ^ is the exponent operator, also referred to as the exponential factorial of 10. It is cited on Frank Pilhofer's Googolplex page as an example of something larger than Googolplexplex.
See also 6pt1.86×10^{3148880079}.
10^{101010103.58259...×103010299956639812}
The value of "latinlatinlatinbyllionyllionyllionyllion" under Knuth's "latin{nameofNwithspacesdeleted}yllion" extension of his yllion naming system, described on pages 310312 of [144]. See also 10^{4194304}.
10^{10101010101.2826×1082} = 6pt1.86×10^{3148880079}
This 2^(3^(4^(5^(6^(7^(8^(9^10))))))), where each ^ is the exponent operator. It is the largest value you can get using one of each of the ten digits 0 through 9, without any symbols or punctuation: 2^{345678910}. (An improvement on Epstein's suggestion^{84} by replacing 8^{910} with the larger 8^{910}). See also 10^{1.0979×1019} and 5pt4.8293×10^{183230}.
10^{101010101010101010} = 9pt10 = 10pt1
A powertower of ten 10's. This is 10^{④}10 using the higher hyper4 operator, or 10↑↑10 using Knuth's uparrow notation.
This is 4$ = 4!^{④}4! = 24↑↑24, or "4 superfactorial". It is a power tower of 24's of height 24. See 10^{10102.069×1036305}.
Jonathan Bowers' giggol, 10^{④}100 = 10↑↑100, a powertower of 10's that is 100 numbers high. See here and here for more, see also 10^{3×103000000000+3}.
Using the 195 digits in Numberphile's can of Numberetti, one can make a power tower 163 numbers high, starting with 2^{22...} at the bottom and ending with ...^{1010100000000000000000} at the top. This is not quite as high as the powertower of 256's used to make Steinhaus's "Mega".
Steinhaus's "Mega"
Using the original Steinhaus notation, the number represented by "2 inside a square" is "triangle(triangle(2))", where "triangle(x)" is x^{x}. Thus, 2 = triangle(2^{2}) = triangle(4) = 4^{4} = 256.
Steinhaus defined the "Mega" as "2 inside a circle". In general, "n inside a circle" is n inside n concentric squares, so "2 inside a circle" is "square(square(2))". We already know that square(2) is 256, so "2 inside a circle" is square(256) or "256". This is 256 inside 256 concentric triangles.
To compute this, we begin with x_{0}=256. Let x_{1} = x_{0} to the power of itself. Then let x_{2} = x_{1} to the power of itself. Continue until reaching the value x_{256}. Using a Hypercalc BASIC program shown here, it's easy to find that the answer, converted to base 10, is a powertower with 255 repetitions of the number 10, and 1.9923739...×10^{619} (which is approximately the logarithm to base 10 of 256^{256257}) at the top.
Later, Moser became involved and generalised the notation to include pentagons, hexagons, etc. The pentagon replaced the old circle, and circles no longer had any meaning. Steinhaus's "Mega" would be represented as "2 inside a pentagon" using the SteinhausMoser notation.
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, proceed to the hyper5 discussion on my large numbers page.
Note. I try to explain things at least a little bit, and to give suitable references. I definitely do not follow my own First Law of Mathematics. If you suggest an improvement for these pages, I'll probably be able to do something to make it better — just let me know (contact links at the bottom of the page).
First page . . . Back to page 21 . . . Forward to page 23 . . . Last page (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.
s.11