Factor of Mersenne Number

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

Then both of these properties apply:
 * $$p \equiv 1 \pmod q$$;
 * $$p \equiv \pm 1 \pmod 8$$.

Proof
Suppose $$p \backslash M_q$$.

Then $$2^q \equiv 1 \pmod p$$, and the order of $$2 \pmod p$$ divides $$q$$ from Integer to Power of Multiple of Order.

By Fermat's Little Theorem, the order of $$2 \pmod p$$ also divides $$p-1$$.

Hence we have $$p - 1 = 2 k q$$ and hence $$p \equiv p \pmod q$$.

We also have, from above:
 * $$2^{\left({p-1}\right)/2} \equiv 2 q k \equiv 1 \pmod p$$

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

Thus it follows from the Second Supplement to the Law of Quadratic Reciprocity that $$p \equiv \pm 1 \pmod 8$$.