Quadratic reciprocity

In mathematics, in number theory, the law of quadratic reciprocity connects the solvability of two related quadratic equations in modular arithmetic. As a consequence, it allows us to determine the solvability of any quadratic equation in modular arithmetic.

Related Topics:
Mathematics - Number theory - Modular arithmetic - Quadratic equation

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It was conjectured by Euler and Legendre and first satisfactorily proven by Gauss. Gauss called it the 'golden theorem' and was so fond of it that he went on to provide more than seven separate proofs over his lifetime.

Related Topics:
Euler - Legendre - Gauss - Proofs

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Suppose p and q are two different odd primes, which means that p and q are congruent either to 1 or to 3 (mod 4). If at least one of them is congruent to 1 mod 4, then the congruence

Related Topics:
Odd - Primes - Congruent - Mod

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:x^2equiv p ({ m mod} q)

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has a solution x if and only if the congruence

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:y^2equiv q ({ m mod} p)

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has a solution y. (The two solutions will in general be different.) On the other hand, if both primes are congruent to 3 modulo 4, then the congruence

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:x^2equiv p ({ m mod} q)

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has a solution x if and only if the congruence

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:y^2equiv q ({ m mod} p)

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does not have a solution y.

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