74. For 2n-1 to be prime, n must be prime.
Because 2n'n"-1 is divisible by 2n'-1 and 2n"-1.
The condition does not suffice :
                     211-1 = 2047 = 23 * 89.
The numbers
            N = 2n-1
where n is prime, are called Mersenne numbers.


  75. There are 55 prime numbers less or equal to 257. Mersenne wrote
that only the 11 following values of n give 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).
()See KRAITCHIK, La mathématique des Jeux(Stevens, Bruxelles, 1930),
p.100; ROUSE-BALL, Récréations mathématiques(Hermann, Paris, 1926),
tome I,p.307.