Mathematical_Recreations-Kraitchik-2e

74 Mathematical Recreatwns
The table below gives all results known up to this time.
n F .. Date Author
0-4 primes 1665 Fermat
5 641·6700417 1732 Euler
6 274177 ·67280421310721 1880 Landry
7 [composite, but the]
1909 {Morehead
8 factors are unknown Western
9 divisible by 37·2'. + 1 1903 Western
11 divisible by 39.213 + 1 and 119·2" + 1 1899 Cunningham
12 divisible by 7·2" + 1, 397·2'· + 1, 973.210 + 1 1879 Pervouchine,
Western
15 divisible by 579·2''' + 1 1925 Kraitchik
18 divisible by 13·2'" + 1 1903 Western
23 divisible by 5·2" + 1 1878 Pervouchine
36 divisible by 5·2'" + 1 1886 Seelhof
38 divisible by 3·24' + 1 1903 Cunningham
73 divisible by 5·27" + 1 1905 Morehead
Quite unexpectedly Gauss found that this question was
connected with the problem of inscribing a polygon in a circle.
Gauss proved that the only regular polygons constructible
with ruler and compass are those for which the number of
sides is m = 2k ·F .. ·F .. , "', where k = 0, 1, 2, ... and the
Fermat numbers are all distinct primes.
The first five prime Fermat numbers are 21 + 1, 22 + 1,
24 + 1, 28 + 1, 216 + 1. If we multiply 1 = 21 - 1 into these
numbers we obtain successively 22 - 1,24 - 1,28 - 1,216 - 1,
2 32 - 1, the latter being the product of the first five prime
Fermat numbers. 2 32 - 1 has the following 32 divisors:
1 257 65,537 16,843,009
3 771 196,611 50,529,027
5 1,285 327,685 84,215,045
15 3,855 983,055 252,645,135
17 4,369 1,114,129 286,331,153
51 13,107 3,342,387 858,993,459
85 21,845 5,570,645 1,431,655,765
255 65,535 16,711,935 4,294,967,295