Gordon Spence
Hockey
The exponents n which give Mersenne primes are 2, 3, 5, 7, 13, 17, 19, 31, ... (sequence A000043 in the OEIS) and the resulting Mersenne primes are 3, 7, 31, 127, 8191, 131071, 524287, 2147483647, ... (sequence A000668 in the OEIS).\n', '
If n is a composite number then so is 2n − 1. (2ab − 1 is divisible by both 2a − 1 and 2b − 1.) This definition is therefore equivalent to the definition as a prime number of the form Mp = 2p − 1 for some prime p.\n', '
More generally, numbers of the form Mn = 2n − 1 without the primality requirement may be called Mersenne numbers. Sometimes, however, Mersenne numbers are defined to have the additional requirement that n be prime.\n', 'The smallest composite Mersenne number with prime exponent n is 211 − 1 = 2047 = 23 × 89.\n', '
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