Notable Properties of Specific Numbers

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10^{8.0723047260281×10¹⁵³} = 4^{4^4⁴}

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}.

10^10¹⁶⁶

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 10^{1.877×10⁵⁴}, 10^{3.79×10²⁸¹} and 10^{5.7×10⁴⁰⁵}.

10^{3.79×10²⁸¹}

A highly-simplified 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}.

10^10³⁷⁵

An estimate of the number of distinct universes in the "string theory landscape", given by [137] in section 5, and assuming that the "maximal number of observable e-folds" is about 290. ⁹⁰

10^{5.7×10⁴⁰⁵}

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 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 recurrance of "our" universe. See also 10^{1.877×10⁵⁴}, 10^10¹⁶⁶, 10^{3.79×10²⁸¹} and 10^{10^{1.51×10^{3883775501690}}}.

10^{(6.6556570552...×10⁶⁶⁸)}

This is the factorial of 10⁶⁶⁶, and is called the leviathan number. 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.

10^{2.62230310839×10¹³²²} = 2^{2⁴³⁸⁷×203}×10^2⁴³⁸⁸

The number of lines of text one Adam Clarkson will owe his high school chemistry teacher on September the 22^nd 2010. The story runs as follows¹⁶:

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". 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. [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 of September was 203 days after the 26^th of February, the lines had been doubled 203 times — on that day the assignment was 100×2²⁰³ ≅ 1.2855504354...×10⁶³ lines, a little over one vigintillion, and due the next day. (This quantity is somewhat larger than the capacity of the known universe, even if all the galaxies were converted into paper and ink.) Each day after that the number is squared, so the assignment on the 18^th of September was (100×2²⁰³)² = 100²×2^2×203, about 1.65×10¹²⁶. The day after that it was 100⁴×2^4×203, or about 2.73×10²⁵², and so forth:

on day:	the assignment was:	due on:	on day:	the assignment was/will be:	due on:
19980226	100	19980227	19980921	2^2⁴×203×10^2⁵	19980922
19980227	200	19980228	19990921	2^{2⁴⁺³⁶⁵×203}×10^{2⁵⁺³⁶⁵}	19990922
19980228	4×100	19980301	20000921	2^{2^369+366×203}×10^{2^370+366}	20000922
19980301	2³×100	19980302	20010921	2^{2^735+365×203}×10^{2^736+365}	20010922
19980302	2⁴×100	19980303	20020921	2^{2^1100+365×203}×10^{2^1101+365}	20020922
19980401	2³⁺³¹×100	19980402	20030921	2^{2^1465+365×203}×10^{2^1466+365}	20030922
19980501	2³⁴⁺³⁰×100	19980502	20040921	2^{2^1830+366×203}×10^{2^1831+366}	20040922
19980601	2⁶⁴⁺³¹×100	19980602	20050921	2^{2^2196+365×203}×10^{2^2197+365}	20050922
19980701	2⁹⁵⁺³⁰×100	19980702	20060921	2^{2^2561+365×203}×10^{2^2562+365}	20060922
19980801	2¹²⁵⁺³¹×100	19980802	20070921	2^{2^2926+365×203}×10^{2^2927+365}	20070922
19980901	2¹⁵⁶⁺³¹×100	19980902	20080921	2^{2^3291+366×203}×10^{2^3292+366}	20080922
19980917	2¹⁸⁷⁺¹⁶×100 = 2²⁰³×100	19980918	20090921	2^{2^3657+365×203}×10^{2^3658+365}	20090922
19980918	2⁴⁰⁶×10⁴	19980919	20100921	2^{2^4022+365×203}×10^{2^4023+365}	20100922
19980919	2^4×203×10^2³	19980920	20110921	2^{2^4387+365×203}×10^{2^4388+365}	20110922
19980920	2^2³×203×10^2⁴	19980921	20120921	2^{2^4752+366×203}×10^{2^4753+366}	20120922

So, on the 22^nd September 2010 the number of lines due will be 2^{2⁴³⁸⁷×203}×10^2⁴³⁸⁸, which is about 10^{(3.48926916371×10¹²¹²)}.

The length of the assignment, as the author notes¹⁷, is "somewhat larger" than the just-mentioned count of the number of ways the universe's history could be shuffled. It is so large that if you tried to write the number of lines as a normal decimal number (that is, without using scientific notation) you'd be writing a number over 10¹³²² digits long, a feat which could not be accomplished even if you could fit a googol digits on each particle in the observable universe.

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.

10^{(2.62086...×10⁶⁹⁸⁹)} = .3^{-(.2^{-(.1^-4)})} = (10/3)^{5^10⁴}

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).

10^{10^10000000} = 10^{10^10⁷}

In section 3 of [137] 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").

10^{3×10^3000000000+3} ≅ 10^{10^{10^9.477121...}}

Jonathan Bowers' gigillion, defined precisely as 10^{3×10^3×10⁹+3}. This is part of a sequence, preceded by megillion = 10^{3×10^3×10⁶+3} and followed by terillion = 10^{3×10^3×10¹²+3}. See also 97pt10^10000000000.

10^{10^10¹⁰} = 10^{10^10000000000} = 10^④4

A power-tower of four 10's, written "10^④4" using my hyper4 notation or "10↑↑4" using Knuth's up-arrow notation. See also 97pt10^10000000000.

10^{1.55×10^{4342944819032}} = e^{e^10¹³}

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

10^{3.5536897484442191...×10^{8852142197543270606106100452735038}} ≅ e^{e^e⁷⁹} ≅ 10^{10^10³⁴}

The original (higher) value of the first (Riemann hypothesis true) Skewes number. It is normally written as "10^{10^10³⁴}". It was later reduced to e^{e^27/4}, 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 and #10^{10^{3.3×10⁹⁶³}}.

10^{10^10¹⁰⁰}-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 2^a5^b where a and b can each be as high as a googol (such numbers are found in sequence A3592). Additional factors of 10^{10^10¹⁰⁰}-1 can be found via Fermat's Little Theorem (see 999999). As a result there are a huge number of known factors of 10^{10^10¹⁰⁰}-1, beginning with: 3, 11, 17, 41, 73, 101, 137, 251, 257, 271, 353, 401, 449, 641, 751, ... [121].

10^{10^10¹⁰⁰}

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=10^googolduplex, googolquadriplex, googolquinplex and so on. See also 200¹⁰⁰ and 10^10¹⁰⁰.

10^{10^10¹²²}

In [126] (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 10^{10^10¹²²}. 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.

10^{10^{3.29994322...×10⁹⁶³}} = e^{e^{e^{e^7.705}}}

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^{10^{10¹⁰⁰⁰}} is given.

See also 288 and 22pt1.84×10³³.

10^{10^{10^{1.0126×10^1656520}}} = 3pt1.0126×10^1656520 = e^{e^{e^{e^{e^{e^e}}}}}

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^{+e^{e^{...^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^{e^0.834032...}, 143 would be e^{e^{e^0.471239...}} 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, 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 "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^{10^{10^{10^10000000}}} = 4pt10000000

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^{10^{10^{10^10⁷}}}.

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, 2×10^{10^10¹⁰}, 2×10^④12, 2×10^④20, ..., where ^④ is the higher hyper4 operator. For higher levels the number is 2×10^④(8n-44).

10^{10^{10^{10^{6.8880×10¹⁴}}}} = 4pt6.8880×10¹⁴ = e^{e^{e^{e^e³⁵}}}

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^{10^{10^{10^10000000}}}) and the largest I've seen apart from Graham's number.

10^{10^{10^{10^{10^{4.8293×10¹⁸³²³⁰}}}}} = 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.