Law of quadratic reciprocity

The Law of Quadratic Reciprocity is a fundamental result in number theory that describes when certain quadratic equations have solutions modulo prime numbers. Specifically, it helps determine whether a number is a "quadratic residue" (a perfect square) modulo different primes. The law states a surprising and elegant relationship between pairs of odd primes: whether one prime is a quadratic residue modulo the other depends on whether the other prime is a quadratic residue modulo the first, with specific adjustments for the primes involved. This principle allows mathematicians to solve complex problems about squares and primes more systematically.