Notable Properties of Specific Numbers (page 19) at MROB
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Notable Properties of Specific Numbers
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1.786266437(26)×1041
This is 2 to the power of the reciprocal fine-structure constant<br>137.0359... using the CODATA 2022 recommended value of<br>the latter. It is a simple use of the popular<br>fine structure constant to produce a value<br>close to the Dirac ratio 1040. See also<br>3.377...×1038.
1.15868...×1042 = 64! /<br>(32!×8!2×2!4×24)
This is (a corrected value for) the number of possible chess<br>positions, originally given by Shannon in the 1950 article<br>"Programming a Computer for Playing Chess." (Phil. Mag. 41, 256-275).<br>The formula is based on the idea that you can theoretically arrange<br>all 32 pieces in any position whatsoever (giving 64!/32!) but that all<br>pawns of a given colour are equivalent (8! for each colour), as is each<br>pair of rooks (22) and each pair of knights (another 22); the<br>bishops are not interchangeable but each has only 32 squares to choose<br>from (24). However, this is inaccurate for a number of reasons.<br>First and most important, a pawn cannot switch columns (ranks), or<br>move past the opposing pawn in its rank, unless it captures. The more<br>captures take place, the more flexibility the pawns have, but that<br>decreases the number of pieces which decreases the number of board<br>positions. Also, the possibility of pawn promotion increases the<br>number of combinations somewhat. A far better estimate is that<br>by John Tromp.
The number of possible chess games is much higher.<br>See also 765 and 2.081681...×10170.
20988936657440586486151264256610222593863921 =<br>(2148+1)/17 ~= 2.098893665744×1043
In July 1951 Ferrier found this 44-digit prime using a<br>mechanical desk calculator. It became the largest-known prime,<br>breaking the record set by Lucas in 1876. This<br>record did not stand long; it was broken by<br>Miller and Wheeler in the same month. 34
63976656348486725806862358322168575784124416 ~=<br>6.397665...×1043
This is 447212, and is "nearly equal" to 398712 + 436512:<br>it is a "near miss" for Fermat's Last Theorem. The numbers appear in<br>the Simpsons episode "The Wizard of Evergreen Terrace". See also<br>8712.
1044
The value of the number called zài in Chinese. See also<br>104096.
393050634124102232869567034555427371542904832 ~=<br>3.9305×1044
This is 141×2141+1, the smallest number of the form<br>n2n+1 that is prime. Cullen (the same one after whom the<br>Cullen numbers are named) investigated numbers of this form<br>in 1905.
824792557184288824246737061810550733633916929 =<br>3×(7×392-1)/2 ≈ 8.247925...×1044
This is a lower bound found by Milton Green for the value of BB(8),<br>where BB(n) is the busy beaver function.
7.40119...×1045 = 7!×36 ×<br>24! × 24!/246 = 7401196841564901869874093974498574336000000000
(The 4x4x4 Rubik's cube)
The number of ways to arrange a<br>4×4×4 Rubik's Cube. The corner cubelets have<br>the same number of combinations as the 2×2×2 cube (see<br>3674160). There are 24 edge pieces, which can be put in<br>any of the 24!≈6.2×1023 permutations. There are 24 centre<br>pieces — these would have 24! permutations, except for the fact that<br>each of the four pieces of a given colour are indistinguishable from<br>each other; so there are 24!/246 combinations for those pieces.
See also 3674160, 4.3252...×1019,<br>2.8287...×1074, 1.5715...×10116,<br>and 1.95005...×10160.
2(1.1×140) ≈ 2.28×1046
Randall Munroe made this estimate of<br>the number of "meaningfully different" 140-character Twitter messages<br>in English, using Shannon's estimate [137] of roughly 1.1 bits of<br>information per letter (Twitter has since increased the Tweet length<br>limit to 280, so it would now be about 5.21×1092).
See also 2.45995..×10200,<br>10800, and 4×102254.
4.519364...×1046
An upper bound on the number of possible chess "diagrams" (a board<br>configuration together with the facts of whose turn it is, who still<br>has the option of castling, and any available en passant capture),<br>computed by John Tromp. This estimate is better than that of<br>Will Entriken and far better than<br>mine.
5.4×1047
The amount of energy (in joules, or kg(m/s)^2) released in the<br>"GW150914" event (merging of two co-orbiting black holes) detected by<br>the LIGO gravitational wave experiment on 2015 Sep 14. This is 3 times<br>the mass of the Sun times the<br>speed of light squared. This amount of energy was<br>released in less than 1 second, with a peak intensity about 50 times<br>as great as all of the stars in the universe combined.
2054221614063184107682218077003539824552559296000...