Prove that if the prime $p= 8k + 1$ then
$$p \mid (2^{\frac{p-1}{2}} - 1) $$
By Proposition (7.15), if $p \equiv 1 \pmod 8$ then 2 is a quadratic residue of $p$. By Euler's Criterion
$$ 2^{\frac{p-1}{2}} \equiv 1 \pmod p $$
That is,
$$p \mid (2^{\frac{p-1}{2}} - 1) $$