TITLE: $2^{251 }-1$ not Mersenne Prime
QUESTION [7 upvotes]: I need to show that $2^{251} - 1$ is not a Mersenne prime. Hard because $251$ is prime. If i can show that a prime $p$ is congruent to $3 \bmod 4$, and $q = 2p + 1$ is a prime, then $2^p$ is congruent to $1 \bmod q$... then I can show that $2^{251} - 1$ is not prime. Having difficulty with the middle part . 
Thank you for aid!

REPLY [9 votes]: You have almost done it. Note that $2(251)+1$ is prime. It follows that $503$ divides your Mersenne number.
Remark: The relevant theorem was not stated precisely in the OP.  Here is a precise statement. Let $q$ be a prime of the form $4k+3$. If $2q+1$ is prime, then $2q+1$ divides $2^q-1$.