Theorem

If $p$ is an odd prime, then $$\left( \frac{2}{p} \right)=(-1)^{\frac{p^{2}-1}{8}}=\begin{cases} 1 & \text{if }p\equiv\pm1\pmod{8} \\ -1 & \text{if }p\equiv\pm 3\pmod{8} \end{cases}$$