Quadratic Reciprocity Relations

$$\left({-1\over p}\right) = (-1)^{(p-1)/2}$$ (1)

$$\left({2\over p}\right) = (-1)^{(p^2-1)/8}$$ (2)

$$\left({q\over p}\right) = \left({p\over q}\right)(-1)^{[(p-1)/2][(q-1)/2]},$$ (3)

where $\left({{\textstyle{p\over q}}}\right)$ is the Legendre Symbol.

See also Quadratic Reciprocity Theorem