Jacobi symbol

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

Let $n$ be an odd positive integer, with prime factorization $p_1{}^{e_1}\mathrm{\cdots }p_k{}^{e_k}$. Let $a\ge 0$ 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.