Legendre symbol

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The Legendre symbol $\scriptstyle \left({\frac {a}{p}}\right)\,$, introduced by Adrien-Marie Legendre in 1798, is a multiplicative arithmetic function that gives 0 if $a$ is divisible by the odd prime $p$, otherwise gives –1 if $a$ is a quadratic nonresidue modulo $p$ and +1 if $a$ is a quadratic residue modulo $p$.

$\left({\frac {a}{p}}\right)={\begin{cases}-1{\text{ if }}a{\text{ is a quadratic nonresidue modulo }}p,\\\;\;\,0{\text{ if }}a\equiv 0{\pmod {p}},\\\;\;\,1{\text{ if }}a{\text{ is a quadratic residue modulo }}p{\text{ and }}a\not \equiv 0{\pmod {p}}.\\\end{cases}}$

Two examples:

$\left({\frac {4}{7}}\right)=1$

$\left({\frac {5}{7}}\right)=-1$

See also

• Jacobi symbol