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 $q \divides M_p$, 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.