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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

$\left(\frac{p}{q}\right) \left(\frac{q}{p}\right) = (-1)^{\frac{p-1}{2}\frac{q-1}{2}}$

for p and q odd prime numbers, and $\left(\frac{p}{q}\right)$ denoting the Legendre symbol.

This law, combined with the properties of the Legendre symbol, means that any Legendre symbol $(a/p)$ can be calculated. This makes it possible to determine, for any quadratic equation, $x^2\equiv a \pmod p$, where p is an odd prime, if 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.<ref>Gauss, DA § 4, arts 107–150</ref> He refers to it as the "fundamental theorem" in the Disquisitiones Arithmeticae and his papers, writing

The fundamental theorem must certainly be regarded as one of the most elegant of its type. (Art. 151)

Privately he referred to it as the "golden theorem."<ref>E.g. in his mathematical diary entry for April 8, 1796 (the date he first proved quadratic reciprocity). See facsimile page from Felix Klein's Development of Mathematics in the 19th century</ref> He published six proofs, and two more were found in his posthumous papers. There are now over 200 published proofs.<ref>See F. Lemmermeyer's chronology and bibliography of proofs in the external references</ref>

The first section of this article gives a special case of quadratic reciprocity that is representative of the general case. The second section gives the formulations of quadratic reciprocity found by Legendre and Gauss.