Prove Proposition (4.24) of Chapter 4.
We remind ourselves of Prove Proposition (4.24).
Let $q$ be an odd prime. Any prime factor $p$ of $M_q = 2^q− 1$ is of the form $p \equiv \pm 1 \pmod 8$.
We start with prime $p$ being a factor of $2^q-1$. This means
$$ 2^q \equiv 1 \pmod p $$
Since $q$ is an odd prime, we can write it as $p=2n+1$, for some positive integer $n$.
$$ 2^{2n+1} \equiv 1 \pmod p $$
Multiplying through by 2 is valid since $p \not \mid 2$,
$$ 2^{2n+1}\times 2 \equiv 2^{2n+2} \equiv (2^{n+1})^2 \equiv 2 \pmod p $$
This means 2 must be a quadratic residue of $p$. By the result we proved in Exercise (7.3).3(i) we conclude that $p \equiv \pm1 \pmod 8$.
And so if prime $p$ is a factor of $M_q=2^q-1$ then $p \equiv \pm1 \pmod 8$.