## ::Quadratic reciprocity

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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

- <math> \left(\frac{p}{q}\right) \left(\frac{q}{p}\right) = (-1)^{\frac{p-1}{2}\frac{q-1}{2}}</math>

for *p* and *q* odd prime numbers, and <math>\left(\frac{p}{q}\right)</math> denoting the Legendre symbol.

This law, combined with the properties of the Legendre symbol, means that any Legendre symbol <math>(a/p)</math> can be calculated. This makes it possible to determine, for any quadratic equation, <math>x^2\equiv a \pmod p</math>, 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 ยง 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.

**Quadratic reciprocity sections**

Intro Motivating example Terminology, data, and two statements of the theorem Connection with cyclotomy History and alternative statements Other rings Higher powers See also Notes References External links

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