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

References

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.