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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. ... Read more