# Math Help - Mersenne Prime Question

1. ## Mersenne Prime Question

Let p be an odd prime of the form p = 4k + 3

a) Prove that if q = 2p + 1 is prime, then q divides the Mersenne number $M_{p} = 2^p - 1$.

b) Use part (a) to prove that $M_{11}$, $M_{23}$, and $M_{251}$ are composite numbers.

2. (b) should be easy enough since I know that everything from (a) must hold and I have specific examples.

I don't really get what I'm supposed to use to prove (a). What does q being prime have to do with it dividing the Mersenne number?

3. a.

If p=4k+3 then q=8k+7

q|M_n , in other words is:

2^{(q-1)/2}=2^p \equiv 1(mod q)

It is actually the same to say that Legendre's symbol (2/q)=1.

{(2/q)=1, when q=1(mod 8) or q=7(mod 8)}