Quadratic reciprocity law

In number theory, the law of quadratic reciprocity is a theorem about modular arithmetic that gives conditions for the solvability of quadratic equations modulo prime numbers. There are a number of equivalent statements of the theorem. One version of the law states that for p and q odd prime numbers,

where

denotes the Legendre symbol.

This law, combined with the properties of the Legendre symbol, means that any Legendre symbol can be calculated. This makes it possible to determine, for any quadratic equation, ${\displaystyle x^{2}\equiv a{\pmod {p}},}$ where p is an odd prime, whether it has a solution. However, it does not provide any help at all for actually finding the solution. The solution can be found using quadratic residues.

The theorem was conjectured by Euler and Legendre and first proved by Gauss. He refers to it as the "fundamental theorem" in the Disquisitiones Arithmeticae and his papers, writing

Privately he referred to it as the "golden theorem." He published six proofs, and two more were found in his posthumous papers. There are now over 200 published proofs.

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