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The Legendre symbol
, introduced by Adrien-Marie Legendre in 1798, is a multiplicative arithmetic function that gives 0 if
is divisible by the odd prime
, otherwise gives –1 if
is a quadratic nonresidue modulo
and +1 if
is a quadratic residue modulo
.
![{\displaystyle \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}}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/98ff53d0e35c2519ccb128f81288bb4ceba76884)
Two examples:
![{\displaystyle \left({\frac {4}{7}}\right)=1}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/e06652262cb7286290cffa5a5904cbeba2ae1335)
![{\displaystyle \left({\frac {5}{7}}\right)=-1}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/a253f1615ad55cf91b759ed515d4eddf6b16dcec)
See also