Notable Properties of Specific Numbers  


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9.2494778×104053945 = 213466917 - 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.

104194304

The value of "vigintyllion" under D.E. Knuth's -yllion naming system See also 5pt3.58259..×103010299956639812.

1.259769×106320429 = 220996011 - 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...×106846168

This is 3315. 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=3315. (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 1043000 by Chen and Wang in 1989, and e3100 by M.-Ch. Liu and T. Wang in 2002.

2.99410453×107235732 = 224036583 - 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.

1.2216464×107816229 = 225964951 - 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.

4.27764198×108107891 = 213466916(213466917-1)

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

3.15416507×109152051 = 230402457-1

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.

1.24575039×109808357 = 232582657 - 1

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.

3.1647031×1012978188 = 243112609 - 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.

4.4823309×1014471464 = 224036582(224036583-1)

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.

5.8188737×1017425169 = 257885161 - 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.

5.007673×1025956376 = 243112608(243112609-1)

(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.

1.4403971939817846×1032322801021073740208 = 2(230-1616)

This is (approximately) the maximum value that can be represented in the floating-point format used by MathematicaTM, the symbolic mathametics 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 3.4028236692093×1038, 1.797693134862×10308, 1.1897314953572318×104932 and 4.26448742×102525222.

1.6929646×1034850339

This is (257885161-1)×257885161-1. 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.

4.281248...×10369693099 = 9↑↑3 = 999 = 9387420489

This is the largest number you can express with just three base-10 digits and possibly some symbols and/or parentheses: 999, 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 9th power of the 9th 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 9th power of the 9th power of 9" would normally be interpreted as (99)9, which has only 78 digits, it is clear from the folowing 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, 1010000000000, 101.0979×1019, and 104.0853×10369693099.

8.808074...×10646456992 = 22147483647-1 = 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.

3.01607×103576838408

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...×104990856845 = 3321 = 310460353203

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 9×104990856844. In my notebook I claimed that this was the value of 32, where 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 32 was 3321 when in fact it is 3320:

32 = 33
= (33)3
= ((33)3)3
= 196833
= (1968319683)19683
= 3(9×19683×19683)
= 33(2+9+9) = 3320

1010000000000 = 101010

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 101010.

See also 525600, 8675309, 1010, 1011, 4.28...×10369693099, and 1010100.

1022650000000 ≈ 275250000000

(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:

The possibility (and unliklihood) 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 unliklihood is used as the basis of arguments against natural abiogenesis (see for example [208]).

However, if we assume a sufficiently large universe (such as is predicted by any of several hyperinflationary models, see 101.877×1054, 101010122, 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 1022000000000 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 [180] and [192] 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 101010122.

Due to the extremely inaccurate guesswork required for such an estimate, this number is probably better stated as being something like 101010.3±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: 101010.3±1 cubic angstroms is so close to 101010.3±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...×10178485291567

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

G0 = G1 = 1
Gn+1 = 1/n × (SUMi=0..nGi2)     (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 43rd term.

1010000000000000000 = 101016

This number is given by [192] (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 1040, 10500, 101.877×1054, 101077, 101082, 1010166, 103.79×10281, 1010375, 105.7×10405, 101010000000, 109.35×101414973347, 101010122, and 10101.51×103883775501690.

1080000000000000000 = 108×1016

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

101000000000000000000 = 101018

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 1010 bits per second.

See also 109.35×101414973347.

106000000000000000000 = 106×1018

In 2003 Y. Cheng showed that there is a prime between every pair of consecutive cubes N3 and (N+1)3 for all values of N less than 102000000000000000000 (or N3 less than 106000000000000000000). Proving this for all integers seems like it ought to be easy, but it isn't. See also 1.3063778838.

101.0979×1019 = 2341

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, 4.281248×10369693099, 108.0723047260281×10153, 10(2.62086×106989) and 6pt1.86×103148880079.

101029

An estimate by Max Tegmark [172] 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 6.32×1028, the (approximate) number of protons, neutrons and electrons in a human being.

See also 1010115.

103.0056206947796095239×1029 = (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

The Gamma function comes up in lots of different places in mathematics, and is defined in terms of an integral10. 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 + SUMn=1...inf [ B2n / (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 B2n is a Bernoulli number.

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

101033

Just after citing the one-in-103000000 odds against a parrot typing a novel, Crandall [158] gives the odds against a beer can spontaneously tipping over, "an event made possible by fundamental quantum fluctuations". See also 101036 and 101042.

101036

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 101042.

104.342944...×1036 = e1037

In [182] (page 12), cosmologist Alan Guth suggests that "each second the number of universes that exist is multiplied" by approximately e1037. See 101.877×1054.

1037218383881977644441306597687849648128 = 107×2122 ≈ 103.7218×1037

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 10421.

5.45431...×105121759971936968187500605462505161634910(5.1217599719369×1037) = 2170141183460469231731687303715884105727-1 = 2(2127-1)-1

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

2.604233075698...×10634704607339355474571695927232512278791 ≈ 10(6.3470460733936×1038) = 272727

This is 272727 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 272727 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 272727 are 03683: 272727 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, 272727 ends in 0403683, 27272727 ends in 7450083, 2727272727 ends in 1242083, 272727272727 ends in 7002083, 27272727272727 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 272727 in groups of 3 the result will also be a multiple of 27.

Did I mention that I like the number 27?

101042

According to Crandall [158], 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 liklihood 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 103000000, 101033 and 101036.

101050

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 1.15x1042 and 10120.

103.6893614831612×1050

The number of lynz on its first anniversary.

101.877...×1054 = (e1037)(1.37×1010)

In [182] (page 12), cosmologist Alan Guth suggests that "each second the number of universes that exist is multiplied" by approximately e1037. 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 e1037 to the power of 4.33×1017, which comes out to about 101.877...×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 101016, 101077, 101082, 1010166, 103.79×10281, 1010375, 105.7×10405, 101010000000, 101010122, and 10101.51×103883775501690.

101077

In section 2 of [192] 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 ec e3N, where c is a constant substantially smaller than eN (not the speed of light and N is the number of e-folds. In this case we have ec e3×60 ≈ 101077. Because e3N is so large c can be ignored, and because N is at best a rough guess, it doesn't even matter that ee180 is actually closer to 106.4683×1077. 121

See also 1040, 10500, 101016, 101.877×1054, 101082, 1010166, 103.79×10281, 1010375, 105.7×10405, 101010000000, 101010122, and 10101.51×103883775501690.

101082

Another estimate of the total number of different universes in the multiverse, given in section 4 of [192]. 121

See also 1040, 10500, 101016, 101.877×1054, 101077, 1010166, 103.79×10281, 1010375, 105.7×10405, 101010000000, 101010122, and 10101.51×103883775501690.

1010100

(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 101010100.

1010100 + 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 [158]. 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 272727. 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! ≈ √G (G/e)G

which (using Hypercalc) produces the value

1.62940433245933737341793465298354217282188842671486623036236119369409220294525046866798544708422... × 10995657055180967481723488710810833949177056029941963334338855462168341353507911292252707750506615682567

Others have computed Googol factorial, including Byron Schmuland [171] and Bob Delaney [203]. 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! ≈ √G (G/e)G (1 + 1/12G)

1010115

An estimate by Max Tegmark [172] 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 10110, the (approximate) number of subatomic particles that could fit in a space the size of the visible universe.

See also 101029.

10(3.4677786443...×10130) = 2786!

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 101000000 and floating-point up to about 10101000000.


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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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