User:Sigmundur/wip/Mersenne primes

Theorems about Mersenne numbers
Theorem: If a and p are natural numbers such that ap − 1 is prime, then a = 2 or p = 1.

Theorem: If 2p − 1 is prime, then p is prime.

Theorem: If p is an odd prime, then any prime q that divides 2p − 1 must be 1 plus a multiple of 2p. This holds even when 2p − 1 is prime.

Theorem: If p is an odd prime, then any prime q that divides $$2^p-1$$ must be congruent to $$\pm 1 \pmod 8$$.

Theorem: A Mersenne prime cannot be a Wieferich prime.

Theorem: A prime number divides at most one prime-exponent Mersenne number