quadratic reciprocity

Etymology
The theorem highlights a particular form of reciprocity in the solvability of the quadratic equation a2 = b in modular arithmetic. It was conjectured by and  and first proved by.

Noun

 * 1)  The mathematical theorem which states that, for given odd prime numbers p and q, the question of whether p is a square modulo q is equivalent to the question of whether q is a square modulo p.

Usage notes
There are several equivalent statements of the theorem. One version states that if p and q are odd prime numbers, $$\left(\frac{p}{q}\right) \left(\frac{q}{p}\right) = (-1)^{\frac{p-1}{2}\frac{q-1}{2}}$$, where $$\left({p \over q}\right)$$ is the Legendre symbol. This equation remains valid if $$\left({p \over q}\right)$$ is interpreted as a Jacobi symbol, in which case p and q are (only) required to be odd positive coprime integers. However, the value of the Jacobi symbol is less informative about whether p is a square modulo q (it can reveal that it is not, but not definitively that it it is).

The equation can be used to simplify calculation of the Legendre / Jacobi symbol.