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!