# Jacobi symbol

The is a generalization of the Legendre symbol to all odd positive integers.

Let $n$ be an odd positive integer, with prime factorization ${p_{1}}^{e_{1}}\cdots{p_{k}}^{e_{k}}$. Let $a\geq 0$ be an integer. The Jacobi symbol $\left(\frac{a}{n}\right)$ is defined to be

 $\left(\frac{a}{n}\right)=\prod_{i=1}^{k}\left(\frac{a}{p_{i}}\right)^{e_{i}}$

where $\left(\frac{a}{p_{i}}\right)$ is the Legendre symbol of $a$ and $p_{i}$.

A further generalization of the Legendre symbol, due to Kronecker, is the Kronecker symbol.

Title Jacobi symbol JacobiSymbol 2013-03-22 12:36:26 2013-03-22 12:36:26 mathwizard (128) mathwizard (128) 9 mathwizard (128) Definition msc 11A07 msc 11A15 LegendreSymbol KroneckerSymbol