Quadratic Reciprocity
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High Quality Content by WIKIPEDIA articles! The law of quadratic reciprocity is a theorem from modular arithmetic, a branch of number theory, which gives conditions for the solvability of quadratic equations modulo prime numbers. There are a number of equivalent statements of the theorem, which consists of two "supplements" and the reciprocity law: Let p, q 2 be two distinct (positive odd) prime numbers. Then (Supplement 1) x2 1 (mod p) is solvable if and only if p 1 (mod 4). (Supplement 2) x2 2 (mod p) is solvable if and only if p ±1 (mod 8). (Quadratic reciprocity) Let q = ±q where the sign…mehr

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High Quality Content by WIKIPEDIA articles! The law of quadratic reciprocity is a theorem from modular arithmetic, a branch of number theory, which gives conditions for the solvability of quadratic equations modulo prime numbers. There are a number of equivalent statements of the theorem, which consists of two "supplements" and the reciprocity law: Let p, q 2 be two distinct (positive odd) prime numbers. Then (Supplement 1) x2 1 (mod p) is solvable if and only if p 1 (mod 4). (Supplement 2) x2 2 (mod p) is solvable if and only if p ±1 (mod 8). (Quadratic reciprocity) Let q = ±q where the sign is plus if q 1 (mod 4) and minus if q 1 (mod 4). (I.e. q = q and q 1 (mod 4).) Then x2 p (mod q) is solvable if and only if x2 q (mod p) is solvable.