Notable Properties of Specific Numbers  


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108.0723047260281×10153 = 4444

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=44/4)×4, etc.) See also 101.0979×1019.

1010166

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 Planck-unit volume of the universe at 10123, the number of particles at 1080 and the universe age at 1041 times an "interval" of 10-24 seconds).39 See also 101.877×1054, 103.79×10281 and 105.7×10405.

103.79×10281

(my "simple alternate universe count")

A highly-simplified formula to compute the number of possible universes. N = ev 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 101.877×1054.

105.0867...×10286

This is ee661, an improved (but erroneous) upper-bound for the π(x) vs. li(x) problem (the higher Skewes number), by Alan Turing in an unpublished manuscript. It was corrected to 102.6654...×10536 by Cohen and Mayhew in 1965.

1010375

An estimate of the number of distinct universes in the "string theory landscape", given by [196] in section 5, and assuming that the "maximal number of observable e-folds" is about 290. 121

See also 1040, 10500, 101016, 101.877×1054, 101077, 101082, 1010166, 103.79×10281, 105.7×10405, 101010000000, 101010122, and 10101.51×103883775501690.

105.7×10405

(my "alternate universe count")

The factorial of the single-perturbation 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 wave-functions 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 101.877×1054, 1010166, 103.79×10281 and 10101.51×103883775501690.

102.6654...×10536

This is ee1236, an improvement on the higher Skewes number (and a correction of Alan Turing's 105.0867...×10286) published by Cohen and Mayhew in 1965.

101.100839892045×10540

The number of Adam Clarkson's lynz as of the 10th of August, 2003 (and due the following day) as referenced on a classmate's blog17.

10(6.6556570552...×10668)

This is the factorial of 10666, and is called the leviathan number by Clifford Pickover [180]. The word leviathan refers to a whale or sea-monster. Biblical references to "leviathan" are all in the Old Testament, although 666 is more commonly associated with the last book of the New Testament.

102.96216823...×101542 = 225118×203×1025119

(the ever-increasing number of Adam Clarkson's "Lynz")

The number of lines of text one Adam Clarkson will owe his (former) high school chemistry teacher on 22nd September 2012. The story runs as follows16:

In February (on the 26th, 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 26th 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 26th 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 100×259 ≈ 5.7646×1019 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 there16:

. . . [On the 17th] 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 17th September was 203 days after the 26th of February, the lines had been doubled 203 times — on that day the assignment was 100×2203 ≈ 1.286×1063 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 18th and each day thereafter, the number was squared, so the assignment on the 18th September was (100×2203)2 = 1002×22×203, about 1.65×10126: in one day they went from a vigintillion to being much larger than a googol. The day after that it was 1004×24×203, or about 2.73×10252, and so forth. Within days they blew past the short- and long-scale 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 milli-millillion, and the various record-size 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 103.689×1050, just over G. H. Hardy's estimate of the number of possible chess games; by 8th August 1999 they surpassed a googolplex. By their second anniversary (26th 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 10160 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 grown17 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 anniversary116 the Lynz were in excess of 10101000, and the "Clarkkkkson" function based on them117 was generally regarded118 to be faster-growing than most anything out there. As the author points out,

[...] with the ticking time-bomb that is [The Lynz] squaring every day, it's got a Gmail-style claim to infinity.

This table shows the calculation day-by-day, then by months, then by years to make the formula clear:

     on day: the assignment was: due on: on day: the assignment was/will be: due on:
19980226 100 19980227          20000921 22369+366×203×102370+366 20000922
19980227 200 19980228 20010921 22735+365×203×102736+365 20010922
19980228 4×100 19980301 20020921 221100+365×203×1021101+365 20020922
19980301 23×100 19980302 20030921 221465+365×203×1021466+365 20030922
19980302 24×100 19980303 20040921 221830+366×203×1021831+366 20040922
19980401 23+31×100 19980402 20050921 222196+365×203×1022197+365 20050922
19980501 234+30×100 19980502 20060921 222561+365×203×1022562+365 20060922
19980601 264+31×100 19980602 20070921 222926+365×203×1022927+365 20070922
19980701 295+30×100 19980702 20080921 223291+366×203×1023292+366 20080922
19980801 2125+31×100 19980802 20090921 223657+365×203×1023658+365 20090922
19980901 2156+31×100 19980902 20100921 224022+365×203×1024023+365 20100922
19980917 2187+16×100 = 2203×102 19980918 20110921 224387+365×203×1024388+365 20110922
19980918 (2203×102)2 = 2406×104 19980919 20120921 224752+366×203×1024753+366 20120922
19980919 24×203×1023 19980920 20130921 225118+365×203×1025119+365 20130922
19980920 223×203×1024 19980921 20140921 225483+365×203×1025484+365 20140922
19980921 224×203×1025 19980922 20150921 225848+365×203×1025849+365 20150922
19990921 224+365×203×1025+365 19990922 20160921 226213+366×203×1026214+366 20160922

So, on the 22nd September 2012 the number of lines due will be 225118×203×1025119, which is about 10(2.96216823×101542).

This real-life story invokes a similar respect for the innumeracy of common people to that described in the ancient chess legend; see also 2.315×1016, 43252003274489856000 and 10137.

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 suggestion, 3.2-(.1-4), was slightly smaller).

101010000000 = 1010107

In section 3 of [196] 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, 103.79×10281, 1010375, 105.7×10405, 101010122, and 10101.51×103883775501690.

104.0853×10369693099

This is the approximate value of 9999. 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 4.281248×10369693099.

109.35...×101414973347

In an answer on Philosophy stackexchange [211], 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 10N 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 261000000000 zeros, and so the number being displayed would be 10261000000000. Rewritten in the form of 1010x, this comes out to about 109.35...×101414973347.

See also 101016 and 101018.

103×103000000000+31010109.477121...

Jonathan Bowers' gigillion, defined precisely as 103×103×109+3. This is part of a sequence, preceded by megillion = 103×103×106+3 and followed by terillion = 103×103×1012+3. See also 97pt1010000000000.

10101010 = 101010000000000 = 104

A power-tower of four 10's, written "104" using my hyper4 notation or "10↑↑4" using Knuth's up-arrow notation. See also 97pt1010000000000.

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

(Skewes' number)

The Skewes' number, in the general sense refers to the lowest value x for which the prime-counting function π(x) is larger than the logarithmic integral function li(x).

eee79 is the original (higher) value of the first (Riemann hypothesis true) Skewes number, published in a 1933 paper. 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, 10103.3×10963 and 4pt6.8880×1014.

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, for the same reason that 1010-1 = 9999999999 = 99×101010101; and because it is divisible by 99, it is divisible by 11. The same principle lets us add factors of 9999 = 104-1, 99999 = 105-1, 99999999=108-1, and any number of the form 102a5b-1 where a and b can each be as high as a googol (such numbers are found in OEIS sequence A3592). 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, ... [176].

101010100

("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 101000000.

A common name for 10googolplex 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=10googolduplex, googolquadriplex, googolquinplex and so on.

See also 200100 and 1010100.

10101.7×10120

In [152] (page 8), Don Page estimates the Poincare recurrence time of a black hole of a mass equivalent to the visible universe to be "101010102.08 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 eN where N is the number of "microstates", and this in turn is equal to ees where s is the maximum possible entropy of the system. s is approximately (r/lp)2 where r is the radius of the universe and lp is the Planck length. Using a current (2014) value for the size of the visible universe we would get (4.45×1026/(1.62×10-35))2 or about 7.5×10122 in the exponent. This number is also described in Numberphile's video The LONGEST Time. See also 10101010101.1.

101010122

In [184] (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, 103.79×10281, 1010375, 105.7×10405, 101010000000, and 10101.51×103883775501690.

10103.29994322...×10963 = eeee7.705

(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 1010101000 is given. It was improved to 102.6654...×10536 by Cohen and Mayhew in 1965, before actual computational results (starting with Lehman's 1.53×101165) took over.

See also 4pt6.8880×1014.

10[10(1.51×103883775501690)] = 10101010101.1

(Don Page's alternate universe count)

This is a quantity of time, estimated by Don Page in [152] (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 101.877×1054, 105.7×10405, and 10101.7×10120.

1010106.8880×1014 = 3pt6.8880×1014 = eeee35

Wolf's incorrect value [206] of the Knapowski number.

1010102.069197...×1036305 = 3pt2.069...×1036305 = 666666

(Pickover's superfactorial of 3)

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

n$ = n!n! = n! ^ n! ^ n! ^ ... ^ n!

where "$" is "superfactorial", 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 and 22pt1.84×1033.

1010101.0126×101656520 = 3pt1.0126×101656520 = eeeeeee

(range of level-index representaton)

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 that for sufficiently high X, 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". The power tower paradox discussion is also relevant.).

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, a "power-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 "3pt1.0126×101656520" 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 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 1.0126×101656520.

1010101010000000 = 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, 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, 2×1012, 2×1020, ..., where is the higher hyper4 operator. For higher levels the number is 2×10(8n-44).

See also 101010106.8880×1014.

101010106.8880×1014 = 4pt6.8880×1014 = eeeee35

The Knapowski number

This number appears in the paper "On sign-changes of the difference π(x) - li x", by S. Knapowski [136]. 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))))/e35.

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 Graham-Rothschild_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 1010101010000000, which is from 1994 and never held the record).

The Knapowski paper is mentioned briefly by Wolf [206] who erroneously says that it is eeee35, a power-tower with one fewer e.

10101010104.8293×10183230 = 5pt4.8293×10183230

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 6pt1.86×103148880079.

10101010103.58259...×103010299956639812

The value of "latinlatinlatinbyllionyllionyllionyllion" under Knuth's "latin{name-of-N-with-spaces-deleted}yllion" extension of his -yllion naming system, described on pages 310-312 of [142]. See also 104194304.

1010101010101.2826×1082 = 6pt1.86×103148880079

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: 2345678910. (An improvement on Epstein's suggestion84 by replacing 8910 with the larger 8910). See also 101.0979×1019 and 5pt4.8293×10183230.

10101010101010101010 = 9pt10 = 10pt1

A power-tower of ten 10's. This is 1010 using the higher hyper4 operator, or 10↑↑10 using Knuth's up-arrow notation.

22pt1.840837...×1033

This is 4$ = 4!4! = 24↑↑24, or "4 superfactorial". It is a power tower of 24's of height 24. See 1010102.069×1036305.

97pt1010000000000 = 100pt1

Jonathan Bowers' giggol, 10100 = 10↑↑100, a power-tower of 10's that is 100 numbers high. See here and here for more, see also 103×103000000000+3.

161pt1017

Using the 195 digits in Numberphile's can of Numberetti, one can make a power tower 163 numbers high, starting with 222... at the bottom and ending with ...1010100000000000000000 at the top. This is not quite as high as the power-tower of 256's used to make Moser's "Mega".

255pt1.9923739...×10619

The "Mega"

This is Leo Moser's "Mega", a power-tower of 256 repetitions of the number 256 (which is itself 44). Converting to base 10, it becomes a power-tower with 255 repetitions of the number 10, and 1.9923739...×10619 (which is approximately the logarithm to base 10 of 256256257) at the top. It is more commonly expressed (using the Steinhaus-Moser notation) as the number 2 inside a square.

 

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).

 


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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.


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