Mersenne numbers

74.For 2Because 2^{n}-1 to be prime, n must be prime.^{n'n"}-1 is divisible by 2^{n'}-1 and 2^{n"}-1. The condition does not suffice : 2^{11}-1 = 2047 = 23 * 89. The numbers N = 2^{n}-1 wherenis prime, are calledMersenne numbers.

75. There are 55 prime numbers less or equal to 257. Mersenne wrote that only the 11 following values ofngive prime numbers : 2, 3, 5, 7, 13, 17, 19, 31, 61, 127, 257. The question of Mersenne numbers is not completely solved(1): a) The primality of N has been verified for 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127. b) N is known composite for 11, 23, 29, 37, 41, 43, 47, 53, 59, 67, 71, 73, 79, 83, 97, 113, 131, 151, 163, 173, 179, 181, 191, 197, 211, 223, 233, 239, 251, 101, 103, 109, 137, 139, 257. (The complete factorisation is known for the numbers of the 1st line; Factorisation is incomplete for the numbers of the 2nd and 3rd lines. As for the numbers of the 4th line, no factor is known.) c) Finally, nothing is proved for 149, 157, 167, 193, 199, 227, 229, 241.

76. Even though the affirmation of Mersenne contains few errors, it is nevertheless troubling. Rouse-Ball wrote : <<Mersenne was a good mathematician, not an exceptional one and it would be strange that he would succeed to establish a proposition where men such as Euler, Lagrange, Legendre, Gauss, Jacobi and other mathematicians of first rank failed. But if the proposition comes from Fermat, with whom Mersenne corresponded regularly, the question is different and not only one explains the lack of demonstration, but also one is not sure the problem has been properly attacked.>> It is difficult to imagine a nicer compliment to Fermat.

77. Mersenne numbers play an essential role in the study of perfect numbers (see XI, D).

(1)See KRAITCHIK, La mathématique des Jeux(Stevens, Bruxelles, 1930), p.100; ROUSE-BALL, Récréations mathématiques(Hermann, Paris, 1926), tome I,p.307.

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