An integer ${\displaystyle \alpha }$ may be called a quadratic residue modulo ${\displaystyle \beta }$ if there exists an integer, ${\displaystyle \gamma }$, such that the congruence,

${\displaystyle \alpha \equiv \gamma ^{2}{\pmod {\beta }}}$

holds. Else ${\displaystyle \alpha }$ is a quadratic nonresidue modulo ${\displaystyle \beta }$.