Legendre symbol

In number theory, the Legendre symbol is a function of a {\displaystyle a} and p {\displaystyle p} defined as ( a p ) = { 1 if a is a quadratic residue modulo p and a ≢ 0 ( mod p ) , − 1 if a is a quadratic nonresidue modulo p , 0 if a ≡ 0 ( mod p ) . {\displaystyle \left({\frac {a}{p}}\right)={\begin{cases}1&{\text{if }}a{\text{ is a quadratic residue modulo }}p{\text{ and }}a\not \equiv 0{\pmod {p}},\\-1&{\text{if }}a{\text{ is a quadratic nonresidue modulo }}p,\\0&{\text{if }}a\equiv 0{\pmod {p}}.\end{cases}}} where p {\displaystyle p} is an odd prime number and a {\displaystyle a} is a positive integer that may or may not be a quadratic residue mod p. The Legendre symbol is a multiplicative function. The Legendre symbol was introduced by Adrien-Marie Legendre in 1797 or 1798 in the course of his attempts at proving the law of quadratic reciprocity. Generalizations of the symbol include the Jacobi symbol and Dirichlet characters of higher order. The notational convenience of the Legendre symbol inspired introduction of several other "symbols" used in algebraic number theory, such as the Hilbert symbol and the Artin symbol.