First I want to define the Legendre symbol $(\frac{a}{p})$.
$(\frac{a}{p}) = 1$ when a is a square mod p, and $(\frac{a}{p}) = -1$ otherwise.
I want to figure out when $(\frac{2}{p}) = 1, -1$. According to the answers, $(\frac{2}{p}) = 1$ for $p = 1, -1 (mod 8)$ and $(\frac{2}{p}) = -1$ for $p = 3, -3 (mod 8)$. What I can't figure out is how to get to the answer.
What I do know is that for any odd prime $p$, $p = 1, 3 (mod 4)$ and intuitively it seems that this is related to $mod 8$ somehow, but I'm not sure how to make the connection.
Any help would be appreciated! (And apologies in advance for my horrible formatting here.)