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Legendre Symbol

\left({m\over n}\right)=(m\vert n)\equiv\cases{ 0 & if $m\ve...
... $m$\cr -1 & if $n$\ is a quadratic nonresidue modulo $m$.\cr}

If $m$ is an Odd Prime, then the Jacobi Symbol reduces to the Legendre symbol. The Legendre symbol obeys $(ab\vert p)=(a\vert p)(b\vert p)$.

\left({3\over p}\right)=\cases{
1 & if $p\equiv \pm 1\ \lef...
-1 & if $p\equiv \pm 5\ \left({{\rm mod\ } {12}}\right)$.\cr}

See also Jacobi Symbol, Kronecker Symbol, Quadratic Reciprocity Theorem


Guy, R. K. ``Quadratic Residues. Schur's Conjecture.'' §F5 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 244-245, 1994.

Shanks, D. Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, pp. 33-34 and 40-42, 1993.

© 1996-9 Eric W. Weisstein