Factor of Mersenne Number equivalent to +-1 mod 8

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Theorem

Let $p$ and $q$ be prime numbers such that $q$ is a divisor of the Mersenne number $M_p$.

Then:

$q \equiv \pm 1 \pmod 8$


Proof

Suppose $q \mathop \backslash M_p$.

From Factor of Mersenne Number $M_p$ is of form $2 k p + 1$:

$q - 1 = 2 k p$


From above:

$2^{\left({q - 1}\right) / 2} \equiv 2 k p \equiv 1 \pmod q$

and so $2$ is a quadratic residue $\pmod q$.

From Second Supplement to Law of Quadratic Reciprocity:

$q \equiv \pm 1 \pmod 8$

$\blacksquare$


Sources