let p and q be prime numbers with p=3 mod 4 and q=2p + 1. Prove that 2^p -1 is a Mersenne prime if and only if p=3.
Since it means thus . Therefore, is a quadradic residue modulo . Thus, there is a solution to . Therefore, . Therefore, we see that divides . However, can still be prime if . However, if then which would mean that is not prime. The only way, therefore, for it to be prime is when .