# Notable Properties of Specific Numbers

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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.

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

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.

This is 3^{315}. In 1937 Vinogradov showed that the weak
Goldbach conjecture is true for all numbers
larger than some V, and in 1956 Borodzkin showed that Vinogradov's
result was true for V=3^{315}. (The weak Goldbach conjecture
states that all odd numbers greater than 7 are the sum of three
primes, for example 11=3+3+5.) This result was later improved to about
10^{43000} by Chen and Wang in 1989, and
e^{3100} by M.-Ch. Liu and T. Wang in 2002.

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.

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.

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

(2013 Mersenne prime record)

Discovered in 2013, 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 Dr. Curtis Cooper and GIMPS project. 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.

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,
Hypercalc achieves a far greater range than any
conceivable floating-point format by representing numbers in a
different way).

See also 3.4028236692093×10^{38},
1.797693134862×10^{308},
1.1897314953572318×10^{4932} and
4.26448742×10^{2525222}.

This is (2^{57885161}-1)×2^{57885161-1}. The number 57885161 is a
Mersenne_prime.

(largest known perfect number in 2013)

As of 2013 Jan 25, this was the largest known perfect number. See here for a complete list of known perfect numbers.

4.281247...×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 (9^{9})^{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.

See also 387420489, 10460353203,
10^{10000000000}, 10^{1.0979×1019},
and 10^{4.0853×10369693099}.

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 genes, and even fewer possibilities in what those genes contain, so a realistic "number of distinct possible human beings" would be much smaller. See Human genome.

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_{⑥}2, 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_{⑥}2 was 3^{321} when in fact it is
3^{320}:

3_{⑥}2 = 3_{⑤}3

= (3_{④}3)_{④}3

= ((3^{3})^{3})_{④}3

= 19683_{④}3

= (19683^{19683})^{19683}

= 3^{(9×19683×19683)}

= 3^{3(2+9+9)} = 3^{320}

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" (10^{10}), 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}.

See also 525600, 8675309,
10^{10}, 10^{11},
0118 999 881 999 119 725 3,
4.28...×10^{369693099}, and
10^{10100}.

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 [213]).

However, if we assume a sufficiently large universe (such as is
predicted by any of several hyperinflationary models, see
10^{1.877×1054},
10^{1010122}, 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 10^{50} 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 [185] and [197] 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^{1010122}.

Due to the extremely inaccurate guesswork required for such an
estimate, this number is probably better stated as being something
like 10^{1010.3±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^{1010.3±1} cubic angstroms is so close to
10^{1010.3±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.

This G_{43}, the first element in the Göbel sequence G_{n} that
is not an integer, where G_{n} is given by:

G_{0} = G_{1} = 1

G_{n+1} = 1/n × (SUM_{i=0..n}G_{i}^{2}) (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^{10000000000000000} = 10^{1016}

This number is given by [197] (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). This
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 certain anthropic explanations of the origin of the
universe and to the interpretation of "multiple-universe" implications
of cosmic inflation models.^{121}

See also 10^{40}, 10^{500},
10^{1.877×1054}, 10^{1077},
10^{1082}, 10^{10166},
10^{3.79×10281}, 10^{10375},
10^{5.7×10405},
10^{1010000000},
10^{9.35×101414973347},
10^{1010122}, and
10^{101.51×103883775501690}.

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
10^{63}.

10^{1000000000000000000} = 10^{1018}

A very rough estimate of the number of possible life-experiences a
person can have (which is not quite the same as this).
This is based on a sensory bandwidth of 10^{10} bits per second.

See also 10^{9.35×101414973347}.

10^{6000000000000000000} = 10^{6×1018}

In 2003 Y. Cheng showed that there is a prime between every pair of
consecutive cubes N^{3} and (N+1)^{3} for all values of N less
than 10^{2000000000000000000} (or N^{3} less than
10^{6000000000000000000}). Proving this for all integers seems like
it ought to be easy, but it isn't. See also 1.3063778838.

The largest number that can be formed from the digits 1, 2, 3 and 4
using the ordinary functions addition, multiplication and/or
exponents. See also 163, 10460353203,
4.28...×10^{369693099},
10^{8.0723047260281×10153},
10^{(2.62086×106989)} and
6pt1.86×10^{3148880079}.

An estimate by Max Tegmark [175] of the distance (in meters) between you and "an identical copy of you", assuming that the universe is "infinite [and] ergodic" due to unending cosmic inflation. (You and the copy cannot see each other because you are well beyond each other's cosmological horizon.)

Note that this is close to 10 to the power of
6.32×10^{28}, the (approximate) number of protons,
neutrons and electrons in a human being.

See also 10^{10115}.

10^{3.005620694779609...×1029} = (27!)!

Very large factorials like this one can be computed with Stirling's series, a more accurate form of the better-known "Stirling's formula". The series gives a value for the logarithm of the Gamma function.

The Gamma function comes up in lots of different places in
mathematics, and is defined in terms of an integral^{10}. For
positive integers, the value of the Gamma function is equal to the
factorial of the integer plus 1.

The Gamma function can be computed by the following series (which gives its logarithm)

ln gamma(z) = 1/2 ln(2 π) + (z + 1/2) ln z - z
+ SUM_{n=1...inf} [ B_{2n} / (2 n (2n-1) z^(2n-1) ) ]

= 1/2 ln(2 π)

+ (z + 1/2) ln z

- z

+ 1/(12 z)

- 1/(360 z^3)

+ 1/(1260 z^5)

- ...

where B_{2n} is a Bernoulli number.

The Barnes' G-function has a similar relationship to the superfactorials, as does the K-function to the hyperfactorials.

Just after citing the one-in-10^{3000000} odds against
a parrot typing a novel, Crandall [160] gives the odds
against a beer can spontaneously tipping over, "an event made
possible by fundamental quantum fluctuations". See also
10^{1036} and 10^{1042}.

The approximate odds against a person living at least 1000 years, as given by life insurance tables quoted by William Feller, in "Probability Theory and its Applications". (The tables don't actually go up that far; they simply give an extrapolation formula for ages above a certain point.)

See also 10^{1042}.

10^{4.342944...×1036} = e^{1037}

In [187] (page 12), cosmologist Alan Guth suggests that "each
second the number of universes that exist is multiplied" by
approximately e^{1037}. See 10^{1.877×1054}.

10^{37218383881977644441306597687849648128} =
10^{7×2122} ≈ 10^{3.7218×1037}

This number is described in the Mahayana Buddhist
scripture Buddha-avatamsaka-nama-vaipulya-sutra (Flower Garland
Sutra of Great Universal Buddha, or simply
Avatamsaka, in book 30, "the Incalculable")
which dates from about 420 CE. In Japanese its name is pronounced
hukasetsuhukasetsuten (ふかせつふかせつてん); one Chinese
pronunciation is bukeshuo bukeshuo zhuan. ^{55},^{56},^{57},^{129}.

See also 10^{421}.

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

This is C_{5} in Catalan's sequence and conjectured
to be prime. It's a little too big to test.

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

This is 27^{2727} calculated to 50 significant digits with
Hypercalc. It has over 10^{38} 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 log_{10}27×27^{27}, 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 = 27^{2803} mod 100000 =
03683.

By extending this method recursively (by the method
described here) it can be shown that 27^{27} ends in
9892803, 27^{2727} ends in 0403683, 27^{272727} ends in
7450083, 27^{27272727} ends in 1242083,
27^{2727272727} ends in 7002083,
27^{272727272727} 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?

According to Crandall [160], 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 ([134] 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}.

An estimate of the number of possible chess games, given by G. H.
Hardy ("Ramanujan: Twelve Lectures on Subjects Suggested by His Life
and Work", 1999). See also 1.15x10^{42} and
10^{120}.

The number of lynz on its first anniversary.

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

In [187] (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 13.7-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
4.33×10^{17}, 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 hypothesized universes. Nevertheless, it shows that current inflationary cosmology provides for a possibility similar to the "alternate universe count" I describe here.

See also 10^{1016}, 10^{1077},
10^{1082}, 10^{10166},
10^{3.79×10281}, 10^{10375},
10^{5.7×10405},
10^{1010000000}, 10^{1010122},
and 10^{101.51×103883775501690}.

In section 2 of [197] the authors 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 e3N},
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 e3×60} ≈ 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}

See also 10^{40}, 10^{500},
10^{1016}, 10^{1.877×1054},
10^{1082}, 10^{10166},
10^{3.79×10281}, 10^{10375},
10^{5.7×10405},
10^{1010000000}, 10^{1010122},
and 10^{101.51×103883775501690}.

Another estimate of the total number of different universes in the
multiverse, given in section 4 of [197]. ^{121}

See also 10^{40}, 10^{500},
10^{1016}, 10^{1.877×1054},
10^{1077}, 10^{10166},
10^{3.79×10281}, 10^{10375},
10^{5.7×10405},
10^{1010000000}, 10^{1010122},
and 10^{101.51×103883775501690}.

(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^{1010100}.

googolplex plus one. This number is known to not be
prime. The smallest known factor is
316912650057057350374175801344000001 = 2^{104}5^{6}+1,
found by Robert J. Harley using modular arithmetic [160].
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 4.57936×10^{917}

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 10^{100}, 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 [174] and Bob Delaney [208]. 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)

An estimate by Max Tegmark [175] 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.

Note that this number is "close" (as close as such things usually
ever get) to 10 to the power of 10^{110}, the
(approximate) number of subatomic particles that could fit in a space
the size of the visible universe.

See also 10^{1029}.

10^{(3.4677786443...×10130)} = 27^{86!}

This is an example of a calculation that can be performed easily
either with Maxima^{5} or Hypercalc. Both
require a special command or syntax to get a full precision exponent.
In Maxima:

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^{101000000}.

First page . . . Back to page 20 . . . Forward to page 22 . . . 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.

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