Jacobi symbol JacobiSymbol 2002-04-22 17:53:47 2004-08-24 16:08:52 Definition % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here The {\bf 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 {\em 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.