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.