Question

Let p be a prime with p ≡ 3 (mod 4), and suppose that q = 2p + 1 is also prime.

(a) Show that 2 is a square mod q.

(b) Show that 2^p ≡ 1 (mod q), so q | j 2^p - 1. (This explains the fact that 23 | 2¹¹ - 1.)

It is not yet proved that there are infinitely many primes p such that 2p + 1 is prime. It is also not yet proved that there are infinitely many primes p for which M_p is composite.