Notable Properties of Specific Numbers

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The "weak" Goldbach conjecture states that all odd numbers greater than 5 are the sum of three primes (allowing a prime to be used twice), for example 7=2+2+3 and 27=3+11+13.

In 1937 Ivan Matveevich Vinogradov showed that it is true for all numbers larger than some V, and in 1956 his student Konstantin Grigoryevich Borozdkin showed that Vinogradov's result was true for V=e^{e^{16.038)), which is about 8.005792×10^4008659. The Wikipedia page approximates this as 3^3¹⁵ ≈ 3.248...×10^6846168

This result was later improved to about 10⁴³⁰⁰⁰ by Chen and Wang in 1989, and e³¹⁰⁰ by M.-Ch. Liu and T. Wang in 2002. The conjecture was proved outright in 2013 by Harald Helfgott, but the proof took several years to gain acceptance and had yet to be published as of 2021.

9.249477...×10^4053945 = 2^13466917 - 1

From late 2001 until 2003 Nov 17 this was the largest known prime number (the current record is here). See this list of all known Mersenne primes.

10^4194304

The value of "vigintyllion" under D.E. Knuth's -yllion naming system See also 5pt3.58259..×10^{3010299956639812}.

8.6...×10^4515449 = (4³⁰⁰)²⁵⁰⁰⁰

An estimate of the number of combinations of DNA base-pairs in human DNA that could affect the active regions (genes), from this article.

It is larger than the likely number of viable genotypes (see 3.98×10⁶⁰²⁰) because of the DNA-related effects that "turn on or off" genes during development and later in life. These come from DNA outside the protein-coding regions, and interactions between one gene's expressed protein and the environment causing the turning on or off of another gene.

It is smaller than the total number of base-pair permutations including non-protein-coding regions (see 3.015...×10^3576838408) because most such variations have one of more essential genes that are broken, i.e. "fatal mutations".

10^6000000

The value of a "milli-million" in the original (long scale) system of Chuquet-like names by W. D. Henkle published in 1860. See 10^3000003 for more.

1.259769×10^6320429 = 2^20996011 - 1

Discovered on 2003 Nov 17, and until 2004 May 15 was the largest known prime number (the current record is here). See this list of all known Mersenne primes.

3.248...×10^6846168

See 8.005792×10^4008659

2.994104...×10^7235732 = 2^24036583 - 1

Discovered on 2004 May 15, and and until 2005 Feb 18 was the largest known prime number (the current record is here). See this list of all known Mersenne primes.

1.221646...×10^7816229 = 2^25964951 - 1

Discovered on 2005 Feb 18, and until 2005 Dec 15 was the largest known prime number (the current record is here). It is a Mersenne prime, discovered by Dr. Martin Nowak, a member of the GIMPS (Great Internet Mersenne Prime Search) project. See this list of all known Mersenne primes.

4.277641...×10^8107891 = 2^13466916(2^13466917-1)

For a while this was the largest known perfect number. The current record is here. See here for a complete list.

3.154164..×10^9152051 = 2^30402457-1

The 43^rd Mersenne prime. Discovered on 2005 Dec 15, and until 2006 Sep 4 was the largest known prime number (the current record is here). See this list of all known Mersenne primes.

1.245750..×10^9808357 = 2^32582657 - 1

The 44^th Mersenne prime. Discovered on 2006 September 4, and for a while held the record for largest known prime number (the current record is here). See this list of all known Mersenne primes.

10^10694986

Found by Samuel Yates sometime between 1987 and 1990 [172], a very large Smith number is (10¹⁰³²-1)×(10⁴⁵⁹⁴+3×10²²⁹⁷+1)¹⁴⁷⁶×10^3913210, which comes out to approximately 10^10694985. The first part (10¹⁰³²-1) is 9 times the largest known repunit prime, R₁₀₃₁; the part in the middle (10⁴⁵⁹⁴+3×10²²⁹⁷+1) is the 1987 Dubner palindromic prime. See also 4937775, 10^13614514, and 10^32066910.

3.16470..×10^12978188 = 2^43112609 - 1

(2008 Mersenne prime record)

Discovered in 2008, and for 4 years was the largest known prime number (the current record is here). See this list of all known Mersenne primes.

10^13614514

Found by Samuel Yates in 1990 [172], a very large Smith number is (10¹⁰³²-1)×(10⁶⁵⁷²+3×10³²⁸⁶+1)¹⁴⁷⁶×10^3913210, which comes out to approximately 10^13614514. The first part (10¹⁰³²-1) is 9 times the largest known repunit prime, R₁₀₃₁; the part in the middle (10⁴⁵⁹⁴+3×10²²⁹⁷+1) is the 1990 Dubner palindromic prime. See also 4937775, 10^10694986, and 10^32066910.

4.482330..×10^14471464 = 2^24036582(2^24036583-1)

For a while this number was the largest known perfect number (The current record is here). See here for a complete list of perfect numbers.

5.818872..×10^17425169 = 2^57885161 - 1

Discovered in 2013, and for a couple years was the largest known prime number (the current record is here). See this list of all known Mersenne primes.

3.003764...×10^22338617 = 2^74207281 - 1

(2015 Mersenne prime record)

Discovered in 2015, and for a few years was the largest known prime number (the current record is here). See this list of all known Mersenne primes.

4.67333...×10^23249424 = 2^77232917 - 1

(early 2018 Mersenne prime record)

Discovered in early 2018, and for most of that year was the largest known prime number (the current record is here). See this list of all known Mersenne primes.

1.48894...×10^24862047 = 2^82589933 - 1

(2018 December Mersenne prime record)

Discovered on 21^st December 2018, and for about six years was the largest known prime number. It is credited to Patrick Laroche and the GIMPS project. (The current largest prime is here; see this list of all known Mersenne primes.)

5.00767..×10^25956376 = 2^43112608(2^43112609-1)

(largest known perfect number in 2008)

In 2008 and 4 following years this was the largest known perfect number (The current record is here). See here for a complete list of perfect numbers.

10^32066910

Found by Patrick Costello in 2002 [172], a very large Smith number is (10¹⁰³²-1)×(10²⁸⁵⁷²+8×10¹⁴²⁸⁶+1)¹⁰²⁷×10^2722434, which comes out to approximately 10^32066910. The first part (10¹⁰³²-1) is 9 times the largest known repunit prime, R₁₀₃₁; the part in the middle (10²⁸⁵⁷²+8×10¹⁴²⁸⁶+1) is the 2001 Heuer palindromic prime. See also 4937775, 10^10694986, 10^13614514, and 10^107060074.

1.692963..×10^34850339 = (2^57885161-1)×2^57885161-1.

(largest known perfect number in 2013)

In 2013 and 2014 this was the largest known perfect number (The current record is here). See here for a complete list of perfect numbers.

8.81694...×10^41024319 = 2^136279841 - 1

(2024 October Mersenne prime record)

Discovered on 21^st October 2024, and currently the largest known prime number and the largest known Mersenne prime. It was discovered by the GIMPS (Great Internet Mersenne Prime Search) project, and is credited to Luke Durant and the GIMPS project. Notably, this is the first Mersenne prime to be found with GPUs, after about 28 years of new record primes being found with home desktop CPUs. See this list of all known Mersenne primes.

4.511299...×10^44677234

This is (2^74207281-1)×2^74207281-1. The number 2^74207281-1 is a Mersenne_prime.

As of 2015 Jan 7, this was the largest known perfect number. See here for a complete list of known perfect numbers.

1.10847...×10^49724094

This is (2^82589933-1)×2^82589933-1. The number 2^82589933-1 is a Mersenne_prime.

As of 2018 Dec 21, this was the largest known perfect number. See here for a complete list of known perfect numbers.

3.88692...×10^82048639

This is (2^136279841-1)×2^136279841-1. The number 2^136279841-1 is a Mersenne_prime.

(largest known perfect number since late 2024)

As of 2024 Oct 21, this was the largest known perfect number. See here for a complete list of known perfect numbers.

10^107060074

This is a very large Smith number: (10¹⁰³²-1)×(10⁶⁹⁸⁸²+3×10³⁴⁹⁴¹+1)¹⁴⁷⁶×10^3913210, which comes out to approximately 10^32066910. The first part (10¹⁰³²-1) is 9 times the largest known repunit prime, R₁₀₃₁; the part in the middle (10⁶⁹⁸⁸²+3×10³⁴⁹⁴¹+1) is the 2002 Heuer palindromic prime. See also 4937775, 10^10694986, 10^13614514, and 10^32066910.

6.89508...×10^121210694 = 3×2^402653211-2 = 3×2^3×2²⁷+27-2 = 3×2^{3×2^3×2³+3+3×2³+3}-1

This is the value of the "base" when the "strong" Goodstein sequence iteration, starting with 4=2^2¹, reaches zero. The Goodstein iteration is also described in the Wikipedia article on Goodstein's theorem. In that article the number appears as 3×2^402653211 - 2 which is the length of the sequence, one less than the highest base. The expansion

3×2^{3×2^3×2³+3+3×2³+3} - 1

shows the "structure" of this number a bit more clearly, expressing in a way that is somewhat reminiscent of the "hereditary" notation used in Goodstein's theorem; and the expansion

3×2³×2^3×2³×2^{3×2³×2^3×2³} - 1

is a rearrangement that shows the relation to the transfinite-indexed functions in the Hardy hierarchy, specifically f_ω²*3(3), and also equal to f_{ω²*2+ω*2+2}(4).

6.89508...×10^121210694 is also the highest base achieved in the "weak" Goodstein sequence iteration starting with 8=2³, in which the exponents are just normal numbers, not expanded into base-k polynomials. As such it is the 8^th term of OEIS sequence A268687.

1.4403971939817846×10^323228010 ≈ 2^1073740208 = 2^{(2³⁰-1616)}

This is (approximately) the maximum value that can be represented in the floating-point format used by Mathematica^TM, the symbolic mathemetics program by Wolfram Research. The format uses a 31-bit exponent field. I know of no standard (IEEE or otherwise) floating-point format that uses a 31-bit exponent. This is also the largest exponent field of any exponent format I have found (however, Wolfram Alpha's "Power of 10 representation" and Hypercalc achieve a far greater range than any conceivable floating-point format by representing numbers in a different way).

4.281247731...×10^369693099 = 9↑↑3 = 9^9⁹ = 9^387420489

This is the largest number you can express with just three base-10 digits and possibly some symbols and/or parentheses: 9^9⁹, or 9^(9^9), etc.

This number is described in the novel Ulysses by James Joyce, who wrote:

Because some years previously in 1886 when occupied with the problem of the quadrature of the circle he had learned of the existence of a number computed to a relative degree of accuracy to be of such magnitude and of so many places, e.g. the 9^th power of the 9^th power of 9, that, the result having been obtained, 33 closely printed volumes of 1000 pages each of innumerable quires and reams of India paper would have to be requisitioned in order to contain the complete tale of its printed integers of units, tens, hundreds, thousands, tens of thousands, hundreds of thousands, millions, tens of millions, hundreds of millions, billions, the nucleus of the nebula of every digit of every series containing succinctly the potentiality of being raised to the utmost kinetic elaboration of any power of any of its powers

Although the passage "the 9^th power of the 9^th power of 9" would normally be interpreted as (9⁹)⁹, which has only 78 digits, it is clear from the following words "33 closely printed volumes of 1000 pages each" that the number Joyce intended is far larger. With 369693100 digits, printed on both sides of the 33×1000 pages, each side of a page would need to be able to hold 5602 digits. Due to its being mentioned in a published work, this number has an entry in the OEIS, A241298 (follow the link to see the first and last 100 digits of 9^9⁹).

1.478489...×10^422328499

See 6.700591...×10⁷⁴.

8.80806...×10^646456992 = 2^2147483647-1 = 2^(2³¹-1)-1

This is one of the largest Mersenne numbers ever tested by the Lucas-Lehmer test; it was found to be composite. If prime, it would have been a double Mersenne prime.

3.015...×10^3576838408 = 4^5941000000

The number of combinations of 5941000000 base-pairs in a hypothetical set of 46 human chromosomes in which any pattern of base-pairs is possible. In reality, there is a lot of repetition in the genome, only about 45 million base-pairs in the protein-coding genes (see Human genome), and even fewer possibilities in what those genes contain, so a realistic "number of distinct possible human beings" would be much smaller; see 3.98×10⁶⁰²⁰. See also 8.6×10^4515449.

1.0621842147...×10^4990856845 = 3^3²¹ = 3^10460353203

In high school, around the same time I was calculating large integers like this, I also made approximations of even larger numbers using logarithms on a calculator. This is the largest one I tried to actually write down in standard scientific notation. Due to the limited accuracy of my calculator, the closest estimate I could get was 9×10^4990856844. In my notebook I claimed that this was the value of 3_⑥2, where _⑥ represents the sixth function in the hyper series according to the lower "left-hand-associative" definition. But, due to an error in my formulas I thought 3_⑥2 was 3^3²¹ when in fact it is 3^3²⁰:

3_⑥2 = 3_⑤3
= (3_④3)_④3
= ((3³)³)_④3
= 19683_④3
= (19683¹⁹⁶⁸³)¹⁹⁶⁸³
= 3^{(9×19683×19683)}
= 3^{3^(2+9+9)} = 3^3²⁰

10^10000000000 = 10^10¹⁰

The largest finite number indirectly referred to in any published music (as far as I know). My Hero, Zero, the Schoolhouse Rock! song about how the digit '0' is used to multiply any number by powers of 10, includes the lines:

Place a zero after one,
and you've got yourself a ten --
see how important that is!
When you run out of digits
you can start all over again --
see how convenient that is!
That's why with only ten digits, including zero,
you can count as high you could ever go --
forever, towards infinity.
No-one ever gets there, but you could try ...
with ten billion zeros.

It doesn't exactly say what is being done with those "ten billion zeros" (10¹⁰), but the picture on-screen during the lines "forever, towards infinity / no-one ever gets there, but you could try" shows a pyramid made up of the numbers 9, 80, 700, 6000, 50000, and so on — the screen ends up filled with small zeros — so I imagine they were implying the idea of writing some (nonzero) digit(s) followed by 10,000,000,000 zeros in a row — and then you'd get at least 10^10¹⁰.

10^22650000000 ≈ 2^75250000000

(size of a universe giving rise to spontaneous life)

An estimate of the volume of the universe (in cubic meters), if one makes the following assumptions:

Not all of the universe can be observed directly (because of cosmic inflation),
Life originated purely by the chance meeting of particles to form a single original bacterium, and
That event has happened only once, and all extant life is the result of it.

The possibility (and unlikelihood) of the spontaneous formation of molecules is an important issue in many abiogenesis theories (which attempt to explain the origin of life without assuming the involvement of a supernatural creator). Very complex structures such as an entire bacterium are almost incomprehensibly unlikely to occur on any single Earth-like planet, and this unlikelihood is used as the basis of arguments against natural abiogenesis (see for example [227]).

However, if we assume a sufficiently large universe (such as is predicted by any of several hyperinflationary models, see 10^{1.877×10⁵⁴}, 10^{10^10¹²²}, etc.) then the odds of spontaneous bacterium formation improve significantly — provided that you only care about it happening somewhere. The fact that we happen to be located on the planet where this unlikely event took place then becomes a simple case of observational selection bias (see Anthropic principle).

A size of about 10^22000000000 is the size necessary to guarantee that each possible chance meeting of 75250000000 particles has occurred somewhere at least once; an additional factor of about 10⁵⁰ ensures that this happens in a hospitable environment (a habitable planet).

This number is based on the possibility of a living cell forming through a thermodynamic coincidence. Complex structures can also appear spontaneously through a quantum-mechanical event called a de Sitter fluctuation, and the possibility of such events is important in arguments such as [197] and [209] that attempt to narrow down the possibilities for how the universe might have begun. de Sitter fluctuations can happen either in "normal" universes like our own, or in vast "empty" universes that are predicted by various string-theory models of the beginning of the universe. The important difference is that in an "empty" universe, any spontaneously-appearing life has no chance of continuing to survive, whereas in a normal universe such life can survive if it happens to occur on a habitable planet. This makes it possible to prove that the theories that predict huge amounts of empty space are unlikely to reflect how our own universe originated. See 10^{10^10¹²²}.

Due to the extremely inaccurate guesswork required for such an estimate, this number is probably better stated as being something like 10^{10^10.3±1}. The ±1 in the top exponent allows for a factor of 10 variation regarding the size of the 10^-12-gram bacterium used as the basis of the calculation — this respects the possibility that such a cell is either not complex enough to seed life, or is more complex than necessary. With the ±1 in the exponent, it no longer matters what units we use: 10^{10^10.3±1} cubic angstroms is so close to 10^{10^10.3±1} cubic parsecs that the error term overwhelms the difference. This example is intermediate between everyday innumeracy cases involving class 2 numbers and the power tower paradox that arises at class 4 and above; see also uncomparable.

5.4093...×10^178485291567

This G₄₃, the first element in the Göbel sequence G_n that is not an integer, where G_n is given by:

G₀ = G₁ = 1
G_n+1 = 1/n × (SUM_i=0..nG_i²) (for n>1)

the sequence starts: 1, 1, 2, 3, 5, 10, 28, 154, 3520, 1551880, 267593772160, 7160642690122633501504, ... (Sloane's A3504). The "1/n" in the formula makes it look like there should be fractional terms, but the sequence doesn't actually have any fractional terms until the 43^rd term.

10^9×10¹⁵

Due almost entirely to the popularity of incremental games there has been great demand for JavaScript libraries that can handle numbers beyond the IEEE limit. The most modest (and efficient) of these replace the exponent with another 64-bit floating-point number (used exclusively as a signed integer). Examples include Decimal.js (2014 April, by MikeMcl [232]) and break_infinity.js (2017 November, by Patashu [237]). However, Aarex Tiaokhiao's logarithmica_numerus_lite.js goes higher; and see 3pt1.0126×10^1656520 for more.

10^{10000000000000000} = 10^10¹⁶

This number is given by Linde and Vanchurin [209] (in section 7) as an estimate of the number of possible information "configurations" of a human brain (which is not quite the same as this). The subject of the paper ("How many universes are in the multiverse?") is broader, but in that section they consider certain anthropic explanations of the origin of the universe and to the interpretation of "multiple-universe" implications of cosmic inflation models.¹²¹. In such arguments the number of brain "configurations" presents a limit on the number of possible universes that can be perceived by human observers contained within them, which can present a limit to the number of universes in a multiverse. Without this "observer limitation", the number rises to 10^{10^10000000}.

10^{80000000000000000} = 10^8×10¹⁶

The highest value defined within the counting system set out by Archimedes in "The Sand Reckoner". See here and here. See also 10⁶³.

π^{π^{π^π}} ≈ 9.080217×10^{666262452970848503}

This is π^{π^{π^π}}, and is large enough that we cannot calculate whether it is close to an integer (as is the case with, for example, e^pi×√163). This number might be an integer (or rational, or something simpler than transcendental) — but though it is possible in theory, it is highly likely this number is actually transcendental. Matt Parker has a nice video about this.

See also 10⁴²¹.

5.45431...×10^{51217599719369681875006054625051616349} ≈ 10^{(5.1217599719369×10³⁷)} = 2^{170141183460469231731687303715884105727}-1 = 2^(2¹²⁷-1)-1

This is C₅ in Catalan's sequence and conjectured to be prime. It's a little too big to test.

10^{2.1485709110445×10³⁸} = 2^1729¹²

See 10^{10^{2.1485709110445×10³⁸}}.

2.604233075698...×10^{634704607339355474571695927232512278791} ≈ 10^{(6.3470460733936×10³⁸)} = 27^27²⁷

This is 27^27²⁷ calculated to 50 significant digits with Hypercalc. It has over 10³⁸ digits, which is enough to pretty much guarantee that we will never find out, for example, whether its digits include a run of 40 consecutive 0's. Nevertheless, it is quite easy to figure out its first and last digits. The initial digits are found using logarithms: The logarithm to base 10 of 27^27²⁷ is log₁₀27×27²⁷, quite easily calculated to 50 decimal places as 634704607339355474571695927232512278791.41567985046... The integer part (to the left of the decimal point) tells us what power of 10 it has, and the fractional part (.41567985046...) tells us that the first few digits are 26042...

Perhaps more surprising, the last digits can be calculated by "modulo arithmetic". Modulo arithmetic exploits repeating patterns such as the alternating 125/625 in successive powers of 5. Modulo arithmetic shows that the last five digits of 27^27²⁷ are 03683: 27^27²⁷ mod 100000 = 27^{(27²⁷ mod 5000)} mod 100000 = 27²⁸⁰³ mod 100000 = 03683.

By extending this method recursively (by the method described here) it can be shown that 27²⁷ ends in 9892803, 27^27²⁷ ends in 0403683, 27^{27^27²⁷} ends in 7450083, 27^{27^{27^27²⁷}} ends in 1242083, 27^{27^{27^{27^27²⁷}}} ends in 7002083, 27^{27^{27^{27^{27^27²⁷}}}} ends in 9802083, and all higher power towers of 27's end in 3802083. Each time you add another 27 to the power tower, another final digit becomes constant.

Also, because 27 is a factor of 999 we know that if we add the digits of 27^27²⁷ in groups of 3 the result will also be a multiple of 27.

Did I mention that I like the number 27?

10^10⁴²

According to Crandall [166], mathematician John Littlewood of Cambridge calculated the probability of a mouse surviving on the surface of the sun for a period of one week, based on the likelihood of a suitable number of random fluctuations (brownian motion or quantum fluctuations) to give it a suitable environment for that period of time. This is like Kasner and Newman's thought experiment ([135] pp. 24-25) in which one imagines the odds of a book jumping up into your hand (which they estimate as "between 1/googol and 1/googolplex").

See also 10^3000000, 10^10³³ and 10^10³⁶.

10^10⁵⁰

An estimate of the number of possible chess games, given by G. H. Hardy [134] ("The number of protons in the universe is about 10⁸⁰ / The number of possible games of chess is much larger, perhaps 10^10⁵⁰."). This is far greater than the modern estimate because in Hardy's time the 50-move (optional declared draw) and 75-move (forced draw) rules did not exist; so only the threefold repetition rule applied.

Note that to reach a total of 10^10⁵⁰, and even with players having as many as 30 choices per move, most of the games comprising this total would be well over 10⁴⁸ moves long, and would still be playing well after the last stars burn out. Players would need to carefully produce most of the possible chess positions no more than twice each. It would take a staggeringly large amount of paper or computer memory just to keep track of which positions have been played.

10^{3.6893614831612×10⁵⁰}

The number of lynz on its first anniversary.

10^{1.877...×10⁵⁴} = (e^10³⁷)^{(1.37×10¹⁰)}

In his paper "Eternal inflation and its implications" [199] (page 12), cosmologist Alan Guth suggests that "each second the number of universes that exist is multiplied" by approximately e^10³⁷. If this has been happening for the entire 13.75-billion-year history of our universe, then the number of universes that have been "formed" by this process during the life of our own universe is e^10³⁷ to the power of 4.33×10¹⁷, which comes out to about 10^{1.877...×10⁵⁴}. (There would of course be more if the process began before our universe was created).

Such a calculation does not actually have much meaning: because of general relativity, the passage of time in one universe is not comparable to the passage of time in the false vacuum that generates all these hypothesised universes. Nevertheless, it shows that current inflationary cosmology provides for a possibility similar to the "alternate universe count" I describe here.

10^10⁷⁷

In section 2 of their paper "How many universes are in the multiverse?" [209] Linde and Vanchurin imagine that our universe came about after 60 "e-folds of the slow-roll inflation", and give this as a rough estimate of the number of "universes with different geometrical properties" which will have been created in such a process. The general form of this number is e^{c e^3N}, where c is a constant substantially smaller than e^N (not the speed of light and N is the number of e-folds. In this case we have e^{c e^3×60} ≈ 10^10⁷⁷. Because e^3N is so large c can be ignored, and because N is at best a rough guess, it doesn't even matter that e^e¹⁸⁰ is actually closer to 10^{6.4683×10⁷⁷}. ¹²¹

10^10⁸²

Another estimate of the total number of different universes in the multiverse, given in section 4 of [209]. ¹²¹

10^10¹⁰⁰

(googolplex)

Main article: Googol and Googolplex

Googolplex, for many people is the largest number with a name. Credit for the invention of the -plex suffix is indeterminate. See also 10^{10^10¹⁰⁰}.

10^10¹⁰⁰ + 1

googolplex plus one. This number is known to not be prime. The smallest known factor is 316912650057057350374175801344000001 = 2¹⁰⁴5⁶+1, found by Robert J. Harley using modular arithmetic [166]. Several other larger prime factors are known. Factors of many numbers of the form googolplex+n for small n are listed here.

Results like these are found using methods similar to those in my description of how to find the last few digits of 27^27²⁷. See also 4.57936×10⁹¹⁷

10^{(9.9565705518098×10¹⁰¹)}

The factorial of a googol, called "googolbang" by some. Notice how this appears to be only "a little larger" than googolplex.

Since it is a huge factorial, this number ends in many zeros. Using a multiple-precision calculator and Stirling's approximation, we can actually compute some of the beginning digits of "googol factorial". Letting G be 10¹⁰⁰, the Stirling formula for G! is:

G! ≈ √2πG (G/e)^G

which (using Hypercalc) produces the value

1.62940433245933737341793465298354217282188842671486623036236119369409220294525046866798544708422... × 10^{995657055180967481723488710810833949177056029941963334338855462168341353507911292252707750506615682567}

Others have computed Googol factorial, including Byron Schmuland [183] and Bob Delaney [221]. If you have a really high-precision calculator and want more than 100 of the initial digits of "googolbang", you can use the two-term Stirling series:

G! ≈ √2πG (G/e)^G (1 + 1/12G)

10^{5.887175...×10¹⁰⁴}

A lower bound on the number of possible Go games, using the Superko rule (which prohibits a repetition of any board position that occurred earlier in the game), computed by Matthieu Walraet in 2016. This is far larger then the number of possible chess games, even by Hardy's estimate made before the adoption of the 50- and 75-move draw rules.

Note that to reach this total, and even with players having as many 19×19=361 choices per move, most of the games comprising this total would be well over 10¹⁰⁴ moves long, and would still be playing well after the heat death of the universe. Players would need to carefully produce most of the 2×10¹⁷⁰ possible board positions at some point in the game, without duplication. It would take a staggeringly large amount of paper or computer memory just to keep track of which positions have been played.

10^10¹¹⁵

An estimate by Max Tegmark [184] of the distance (in meters) between us and a "Hubble volume" that contains an indistinguishable copy of our own visible universe, assuming that the universe is "infinite [and] ergodic" due to unending cosmic inflation. We cannot see the denizens of that Hubble volume, and they can't see us, because we are both well beyond each other's event horizons. Any hypothetical "travel" over such a distance would never get there because the cosmic inflation is fast enough to ensure that our destination is moving away from us too fast to reach. Thus, any traveler wishing to make the trip to the identical universe would need to find a way to violate relativistic causality (travelling outside their own Light cone), and perform other surreal feats such as quickly identifying when they have reached the indistinguishable copy and not merely one of the many other possible universes.

Note that this number is "close" (as close as such things usually ever get) to 10 to the power of 10¹¹⁰, the (approximate) number of subatomic particles that could fit in a space the size of the visible universe.

See also 10^10²⁹ and 10^{10^10⁵⁶}.

10^{(3.4677786443...×10¹³⁰)} = 27^86!

This is an example of a calculation that can be performed easily either with Maxima⁵ or Hypercalc. Both require a special command or syntax to get a full precision exponent. In Maxima:

(%i1) fpprec:200; (%o1) 200 (%i2) bfloat(27)^bfloat(86!); (%o2) 9.280229734930337461606281538723272714819546428444775230280\ 92521221347000632381451812127561138285074397038054986794948\ 96418628777914662492770386296874660782494654863382353322231\ 879854474444076252596714 b 34677786443012627135962232742326\ 49403369243699465867529793329082954277942846467082832165998\ 5543139375621253417161850434734823447802

In Hypercalc:

C1 = 27^(86!) R1 = 10 ^ ( 3.4677786443013 x 10 ^ 130 )

Maxima, which is based on the original MIT MACSYMA, performs many of the same functions as the commercial programs Maple, Mathematica and MATLAB, but is free and open-source. It can do exact integer calculations up to about 10^1000000 and floating-point up to about 10^{10^1000000}.

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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 10^9152051 10^10³⁶ 10^{10^10¹⁰⁰} — — footnotes Also, check out my large numbers and integer sequences pages.

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