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 ${p_1}^{e_1} \cdots {p_k}^{e_k}$ . 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.