Mathworld says that $-2$ is a quadratic residue modulo a prime $p$ if and only if $p=8n+1$ or $p=8n+3$, though I don't understand their explanation.
I have seen elementary proofs that $-1$ is a quadratic residue if and only if $p=8n+1$, and $2$ is a quadratic residue if and only if $p = 8n+1$ or $p=8n-1$, but I cannot find (or come up with) a proof for 2. Is there some way to combine the results of $-1$ and $2$, or is there a completely separate way?
Much appreciation, thanks.
Yes, the results can be combined, and you can do it yourself. Just use the general fact that (i) the product of two residues, or two non-residues, is a residue, and (ii) the product of a residue and a non-residue, is a non-residue.
Comment: One way to identify the primes for which $2$ is a quadratic residue is to use Euler's Criterion. The Euler Criterion can also be used to deal directly with $-2$.