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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 .

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    By Fermat's Theorem, $q\equiv 1 \pmod{p}$. If further we ask that $q\equiv 1\pmod 8$, then $q\equiv 1 \pmod {8p}$.2011-11-27
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    @AndréNicolas,you have small typo...it should be $q^{p-1} \equiv 1 \pmod p $2011-11-28
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    It is true that $q^{p-1}\equiv 1 \pmod p$. But I do mean $q\equiv 1 \pmod {p}$. Since $q$ divides $2^p-1$, we know that $2^p\equiv 1 \pmod q$. By Fermat's theorem, $2^{q-1}\equiv 1 \pmod{q}$. Since the order of $2$ modulo $q$ is not $1$, it follows that this order is $p$, and therefore $p$ divides $q-1$. Or in other words, $q \equiv 1 \pmod{p}$.2011-11-28

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