TITLE: Primes that divide $2^{\frac{p-1}2}+1$
QUESTION [5 upvotes]: $p$ is a prime and $p\equiv 3,5 \pmod8$. Prove that $p\ | \ 2^{\frac{p-1}2} +1$.
How should I approach this problem?

REPLY [2 votes]: We're aiming to show then that  $2^\frac{p-1}{2} \equiv -1 \bmod p$
This is equivalent to saying that $2$ is not a quadratic residue $\bmod p$, since if there were some $b$ such that $b^2 \equiv 2 \bmod p$ then since $b^{p-1} \equiv 1 \bmod p$ then also $2^\frac{p-1}{2} \equiv 1 \bmod p$
So, if allowed, you could use the Legendre symbol rules of quadratic reciprocity to complete the proof. Or if not, your result would effectively prove that part of the rules.