Let us denote a prime factor of Mersenne number as $q$ . How to prove following :
$(M_p\equiv0\pmod q \land q\equiv 1 \pmod 8) \Rightarrow q\equiv 1 \pmod {4\cdot p}$
There is a proof that any prime $q$ that divides $2^p − 1$ must be $1$ plus a multiple of $2p$ but I think that is possible to prove this stricter condition. I guess that one should use Fermat's Little Theorem as starting point for this proof .