Factor of Mersenne Number equivalent to +-1 mod 8

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 $q \divides M_p$, where $\divides$ denotes divisibility.

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

$q - 1 = 2 k p$

From above:

$2^{\paren {q - 1} / 2} \equiv 2 k p \equiv 1 \pmod q$

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

• Proof courtesy of The Prime Pages: Modular restrictions on Mersenne divisors.