Legendre symbol

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The Legendre symbol $\left(\frac{a}{p}\right)$, 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 $p$.

$\left(\frac{a}{p}\right)=\{\begin{array}{l}-1\text{ if }a\text{ is a quadratic nonresidue modulo }p,\\ 0\text{ if }a\equiv 0(\mathrm{mod}p),\\ 1\text{ if }a\text{ is a quadratic residue modulo }p\text{ and }a\equiv ̸0(\mathrm{mod}p).\\ \end{array}$

Two examples:

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

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

See also

• Jacobi symbol