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170141183460469231731687303715884105727 = 2127-1 ≅ 1.7014...×1038
Catalan's sequence
This is a Mersenne prime, and the value of C4 in Catalan's sequence given by C0 = 2; Cn+1 = 2Cn - 1. The sequence starts: 2, 3, 7, 127, and then jumps to this value. All five are prime, and Catalan was conjecturing that all Cn are prime.
The primeness of this number was verified by Lucas in 1876. It was the largest prime ever found by hand calculations, and for 75 years was the largest known prime. This record was beaten in 1951 by Ferrier, who used a mechanical desk calculator; subsequent records have been set by electronic computer. 34
It is not known whether C5=2C4-1 is prime.
A Mersenne prime of the form 2M - 1 where M is also a Mersenne prime is called a "double Mersenne prime". This is the largest known double Mersenne prime. The next four candidates are 2213-1-1, 2217-1-1, 2219-1-1 and 2231-1-1, and are all known to be composite.
170141183460469231731687303715884105864 = 2127+136
If you add the numbers in the Catalan sequence 3, 7, 127, 170141183460469231731687303715884105727, you get 3, 10, 137, 2127+136. Since the last two are sort of close to two physical constants, some physics theorists (for example James Gilson, H. Pierre Noyes, Michael Manthey) believe they are part of a "combinatorial hierarchy" that can be used to generate most or all of the physical constants and properties of fundamental particles.
This is (approximately) the maximum value that can be represented in the commonly-used IEEE 754 single-precision (1+8+23 bit) floating-point format.
See also 1.797693134862×10308, 1.1897314953572318×104932, 4.26448742×102525222 and 1.4403971939817846×10323228010.
Physicist Paul Dirac's estimate70 of the ratio of the universe's size to that of a proton. Using present-day values (see 1.73×10-15, 299792458 and 13.73×109) and a naive assumption that the universe grows at the speed of light, the number would be about 7×1039. Dirac noted that it was "close" to the ratio between the strength of gravity and electical attraction between a proton and an electron (about 4.4×1040). He hypothesized that it is more than just coincidence, and proposed that the strength of gravity diminishes with time such that the two numbers remain the same. That would mean that when gravity and electricity were of equal strength the universe was about the size of a proton. The choice of the proton's radius is a bit arbitrary; compare to 8.02×1060, and see also 3.377×1038.
Ratio between the strength of the gravitational and electric attraction of a proton and an electron. This number was considered significant by Dirac, see 1040. Dirac put forth this hypothesis at a time when the proton was still considered fundamental (long before the quark model); note that for other similar pairs of particles (e.g. a proton and a muon, or a positron and an electron) you get different ratios. See also 3.377×1038 and 1.786×1041.
This is 2137.035999084, a simple use of the popular fine structure constant to produce a value close to the Dirac ratio 1040. See also 3.377×1038.
1.15868...×1042 = 64! / (32!×8!2×2!4×24)
This is (a corrected value for) the number of possible chess positions, originally given by Shannon in the 1950 article "Programming a Computer for Playing Chess." (Phil. Mag. 41, 256-275). The formula is based on the idea that you can theoretically arrange all 32 pieces in any position whatsoever (giving 64!/32!) but that all pawns of a given color are equivalent (8! for each color), as is each pair of rooks (22) and each pair of knights (another 22); the bishops are not interchangable but each has only 32 squares to choose from (24). However, this is inaccurate for a number of reasons. First and most important, a pawn cannot switch columns (ranks), or move past the opposing pawn in its rank unless it captures. The more captures take place, the more flexibility the pawns have, but that decreases the number of pieces which decreases the number of board positions. Also, the possibility of pawn promotion increases the number of combinations somewhat. See also this estimate.
The number of possible chess games is much higher. See also 4.63×10170.
20988936657440586486151264256610222593863921 = (2148+1)/17 ~= 2.098893665744×1043
In July 1951 Ferrier found this 44-digit prime using a mechanical desk calculator. It became the largest-known prime, breaking the record set by Lucas in 1876. This record did not stand long; it was broken by Miller and Wheeler in the same month. 34
The value of the number called zài in Chinese. See also 104096.
393050634124102232869567034555427371542904832 ~= 3.9305×1044
This is 141×2141+1, the smallest number of the form n2n+1 that is prime. Cullen (the same one after whom the Cullen numbers are named) investigated numbers of this form in 1905.
824792557184288824246737061810550733633916929 = 3×(7×392-1)/2 ≅ 8.247925...×1044
This is a lower bound found by Milton Green for the value of BB(8), where BB(n) is the busy beaver function.
7.4011968415649×1045 = 7!×36 × 24! × 24!/246
The number of ways to arrange a 4×4×4 Rubik's Cube. The corner cubelets have the same number of combinations as the 2×2×2 cube (see 3674160). There are 24 edge pieces, which can be put in any of the 24!≅6.2×1023 permutations. There are 24 center pieces — these would have 24! permutations, except for the fact that each of the four pieces of a given color are indistinguishable from each other; so there are 24!/246 combinations for those pieces.
See also 3674160, 4.3252×1019, 2.8287×1074, 1.5715×10116, and 1.9501×10160.
2054221614063184107682218077003539824552559296000 = 29×35×53×72×112×132×172×19×23×29×31×37×41×43×47×53×59×61×79×83×89×97×101 ≅ 2.054×1048
The smallest number that has at least 1010 distinct factors. See also 12, 840, 45360, 720720, 3603600, 245044800, 278914005382139703576000 and 457936×10917.
This is an upper bound on the number of possible chess positions, by my reckoning. It allows between 2 and 32 pieces in play, with no more than 16 of one color, including exactly one king of each color, and up to 8 pawns of each color (any of which might have been promoted to another piece). It is higher than Shannon's estimate because it allows pawn promotion.
The number of possible chess games is much higher. See also 8.065817...×1067 and 4.63×10170.
808017424794512875886459904961710757005754368000000000 = 246 × 320 × 59 × 76 × 112 × 133 × 17 × 19 × 23 × 29 × 31 × 41 × 47 × 59 × 71 ≅ 8.08...×1053
This is the "order" (number of elements) in the largest sporadic finite simple group, called the "Monster group" or the Fischer-Griess group.
(Some background: A "group" can be visualized as a set of transformations, e.g. rotations and reflections, that belong to an N-dimensional geometric structure such as a crystal lattice, or Rubik's Cube. A "simple" group has no "subgroups", which are subsets that themselves form a group; a "sporadic" group is one that does not fit into one of the infinite classes (cyclic, alternating, and Lie).)
See also 196883.
Another large number that appears in the Lotus sutra texts of Mahayana Buddhism, where it appears as the word A-so-gi (あそぎ). See also 1011.
A very poor estimate of the radius of the "visible" universe in Planck units, based on its age and the idea that the objects currently at the "edge" have spent half of this time travelling to where they currently appear to be, and then their light spent an equal amount of time coming back to us. More adjustments would need to be made to account for changes in the rate of the universe's expansion, and the amount of its "curvature".
An approximate value for the age of the universe in Planck time units:
r = 13.73×109 × 365 × 24 × 3600 / 5.39 × 10-44
= 8.02 × 1060
For various reasons, this number is not equal to the "radius", nor is it exactly twice the radius of the visible universe. However, for rough calculations of things like the current volume and space-time volume, and particularly for larger derived values like the number of alternate universes, it is more than adequate.
See also 1040.
Archimedes, in his writing psammites (better known as The Sand Reckoner), estimated the size of the universe according to the heliocentric model of Artistarchus, and how many grains of sand would fit in it. He arrived at a value equivalent to one vigintillion, or 1063. Even more impressive, he described a system of numbers extending as high as 108×1016.
H. P. Lovecraft used this number in one of his Cthulhu stories. (See also 1027)
(Personal: For a while during 3rd grade this was the largest number I knew and on a few occasions I wrote it in the sand during recess: 1,000,000,000...,000,000. (counting out 21 sets of zeros). A mean kid would follow and wipe it out.)
The number of ways to shuffle a deck of 52 distinct playing cards. See also 5.23198...×1049.
1.0066961655...×1068 = 20!×319 × 30!×229 / 4
This is the number of ways to arrange the pieces on a Megaminx, a dodecahedral puzzle similar in concept to Rubik's Cube. The reasons for the number are similar to those for the 3x3x3 Rubik's cube except for the parity factor, which is the division by 4. On the Megaminx, every turn performs an even permutation on the corners and an even permutation on the edges. Therefore, the total permutation of all the corners is always even, and likewise for the edges.
See also 9.1197×10262, 1.7989×10571, and 7.7263×10992.
2.8287094227774×1074 = 8!×37 × 12!×210 × 24! × 24!/246 × 24!/246
The number of ways to arrange a 5×5×5 Rubik's Cube. The center cubelets are assumed to be stationary. The 8 corners and the 12 central edge pieces together combine for the same number of combinations as the 3×3×3 Rubik's Cube (see 4.3252×1019). There are another 24 edge pieces, which can be freely placed into any of 24!≅6.2×1023 permutations. There are 48 movable center pieces, in two groups (the ones closer to the corners, and the ones closer to the edge-centers). Each of these two groups of 24 has 24!/246 arrangements for the same reason as the group of 24 center pieces of the 4×4×4 cube (see 7.4012×1045).
See also 3674160, 4.3252×1019, 7.4012×1045, 1.5715×10116, and 1.9501×10160.
5.2106440156792×1078 = 180×(2127-1)2+1
This is a prime, found by Miller and Wheeler in July 1951. This discovery has the distinction of being the first time the record for largest known prime was set by electronic computer. It broke the record set by Ferrier and was soon broken by Robinson. 34
31495448272550005155211307922363110936089435829054233418732462850152371262062592 = 2×136×2256 = 2×136×2223 = 17×2260 ≅ 3.149544...×1079
This is the Eddington number. According to Arthur Eddington in his book Mathematical Theory of Relativity (1923, London, Cambridge University press), it is the number of particles in the universe. It is notable for being the largest specific integer ever thought to have a unique and tangible relationship to the physical world. (All larger numbers in physics are estimates and approximations.)
Eddington was interested in showing that the various physical constants (the speed of light, the gravitational constant, the mass of the electron, etc.) were not accidental but were determined in some way that could be computed exactly. One of these constants was the fine-structure constant
In 1923 the fine-structure constant was known poorly enough that one could surmise that it is exactly 1/136. Eddington computed the number of particles in the universe from other measurements and observations and then found a simple mathematical formula based on integers that gave the same value. (When the fine-structure constant was later found to be closer to 1/137, Eddington repeated his work to make it fit that value. This hurt Eddington's reputation as a scientist, and he was jokingly called "Arthur Adding-one" by detractors.)
Eddington was shown to be wrong on other points. Many other estimates of the number of particles in the universe have been computed, all in the range from 1078 to 1080. Here is an example (which is way too simplistic for cosmologists but shows that the Eddington number was fairly close):
r = radius of visible universe
= age of universe × speed of light
= speed of light / Hubble constant
= 1.42 × 1026 meters
volume of universe = 4/3 π r3
= 1.2 × 1079 cubic meters
average density of universe
= 3 hydroden atoms per cubic meter
(from models that give the minimum mass of a "closed" universe)
1 hydrogen atom = 4 particles (proton + electron, a proton is 3 quarks)
number of particles = 4.8 × 1079
If you include the various massless particles (photons, gravitons, other gauge particles, perhaps neutrinos) and virtual particles, the estimates become much greater. The only estimate I have been able to find gives the density of neutrinos in the cosmic background radiation as being 107 per cubic meter, which gives a value of 1.2×1086 particles in the universe assuming that neutrinos comprise the vast majority.
The Eddington number is approximately the square of Dirac's 1040; see also 3.377×1038.
2350988701644575015937473074444491355637331113544175043017503412556834518909454345703125 = 553 ≅ 2.3509887016443...×1087
In high school I wrote a special program (I think it was on a TRS-80) to calculate and display exact values of large exponents. Then I created tables of powerlogs, (which are integers of the form AAB where A and B are also integers), written by hand in a notebook. This is the largest of about 30 really big numbers in that table. See also 1.0621842147×104990856845.
Some of the larger estimates of the number of particles in the known visible universe are around this value, and result from including photons and other massless particles. The actual number of particles in the universe may be much larger — for example, it might be that most of the universe is beyond our event horizon (redshift horizon). See also 7×1022.
Most pocket calculators max out at 9.9999999...×1099, which is just below a googol (10100, see next entry). See also 9.9999999...×10999 and the computer overflow values starting with 3.4028236692093×1038.
Main article: Googol and Googolplex
10100, which can be called "10 duotrigintillion" but it is better known by the name googol. It can also be expressed 10102, (1010)10 or 10④3 using the lower hyper4 operator.
The dimensionless entropy of a black hole whose mass is about 3.087x1011 solar masses (roughly half the mass of the visible matter in our Milky Way galaxy) would be about 10100. See googolplex and 27.
An estimate of the number of subatomic particles that it would take to fill all the space in the universe. See also 8.45×10184. (From Straight Dope)
1.5715285840103×10116 = 7! × 36 × 24!2 × (24!/4!6)4
The number of ways to arrange a 6×6×6 Rubik's Cube. The corner cubelets have the same number of combinations as the 2×2×2 cube (see 3674160). There are two groups of 24 edge pieces, and 4 groups of 24 center pieces. Within each of these groups of 24 there are 24!≅6.2×1023 arrangements. The center piece groups each have 4!6 fewer combinations because they are in 6 different colors (with 4 of each color) and any pieces of the same color are indistinguishable from each other.
This is over 10 million billion googol. Notice that it is only a little less than the number of ways to play a game of chess, and far greater than the number of valid chessboard positions. When making a move, you have 3×6=18 choices of what to turn and 3 choices of how far to turn it, for a total of 54 choices. This means that in order to get a reasonably good coverage of all the possible combinations, you have to make at least 67 moves to scramble the cube. By contrast, a normal 3×3×3 Rubik's Cube can be scrambled in about 15 moves.
See also 3674160, 4.3252×1019, 7.4012×1045, 2.8287×1074, and 1.9501×10160.
Shannon's estimate of the number of chess games from his original 1950 paper on the topic. It is calculated by the approximation 100040, based on the idea that at each move by White there are about 30 choices, to which Black has about 30 responses, and typical games last 40 moves. See also 1.15x1042 and 101050.
In India's ancient writings there are many references to large numbers with names; some are hard to attach to a specific value because of multiple conflicting or ambiguous uses. One of the larger numbers given a name in India is asankhyeya, commonly said to be 10140. See also 10421 and 103.7218×1037.
1.9500551183732×10160 = 8!×37 × 12!×210 × 24!2 × (24!/246)6
The number of ways to arrange a 7×7×7 Rubik's Cube. The center cubelets are assumed to be stationary. The 8 corners and the 12 central edge pieces together combine for the same number of combinations as the 3×3×3 Rubik's Cube (see 4.3252×1019). There are two additional sets of edge pieces with 24 in each set, which can be freely placed into any of 24!≅6.2×1023 permutations. There are 144 movable center pieces, in six groups (according to where they are in relation to the corners, and the edge-centers). Each of these six groups of 24 has 24!/246 arrangements for the same reason as the group of 24 center pieces of the 4×4×4 cube (see 7.4012×1045).
See also 3674160, 4.3252×1019, 7.4012×1045, 2.8287×1074, and 1.5715×10116.
6.8647976601306×10156 = 2521-1
This is the 13th Mersenne Prime and the first to be found by electronic computer. It was discovered in 1952 by Robinson and breaks the record set by Lucas in 1876, although that record was also broken by the non-Mersenne primes (2148+1)/17 and 180×(2127-1)2+1, which were found the year before. 34
This is (approximately) the number of possible positions in Go, played on a 19×19 board, as given by M. Beeler in HAKMEM72 item 96. See also 1.15x1042.
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