GoKawiilNotable Properties of Specific Numbers
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1.786266437(26)×1041
This is 2 to the power of the reciprocal fine-structure constant 137.0359... using the CODATA 2022 recommended value of the latter. It is 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 colour are equivalent (8! for each colour), as is each pair of rooks (22) and each pair of knights (another 22); the bishops are not interchangeable 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. A far better estimate is that by John Tromp.
The number of possible chess games is much higher. See also 765 and 2.081681...×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
63976656348486725806862358322168575784124416 ~= 6.397665...×1043
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