Mersenne conjectures
In mathematics, the Mersenne conjectures concern the characterization of prime numbers of a form called Mersenne primes, meaning prime numbers that are a power of two minus one.
The original, called Mersenne's conjecture, was a statement by Marin Mersenne in his Cogitata Physica-Mathematica (1644; see e.g. Dickson 1919) that the numbers were prime for n = 2, 3, 5, 7, 13, 17, 19, 31, 67, 127 and 257, and were composite for all other positive integers n ≤ 257. Due to the size of these numbers, Mersenne did not and could not test all of them, nor could his peers in the 17th century. It was eventually determined, after three centuries and the availability of new techniques such as the Lucas–Lehmer test, that Mersenne's conjecture contained five errors, namely two are composite (n = 67, 257) and three omitted primes (n = 61, 89, 107). The correct list is: n = 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107 and 127.
While Mersenne's original conjecture is false, it has led to the New Mersenne conjecture and the Lenstra–Pomerance–Wagstaff conjecture.
The New Mersenne conjecture or Bateman, Selfridge and Wagstaff conjecture (Bateman et al. 1989) states that for any odd natural number p, if any two of the following conditions hold, then so does the third:
If p is an odd composite number, then 2p − 1 and (2p + 1)/3 are both composite. Therefore it is only necessary to test primes to verify the truth of the conjecture.
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