# 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^{}.

Title | Jacobi symbol |
---|---|

Canonical name | JacobiSymbol |

Date of creation | 2013-03-22 12:36:26 |

Last modified on | 2013-03-22 12:36:26 |

Owner | mathwizard (128) |

Last modified by | mathwizard (128) |

Numerical id | 9 |

Author | mathwizard (128) |

Entry type | Definition |

Classification | msc 11A07 |

Classification | msc 11A15 |

Related topic | LegendreSymbol |

Related topic | KroneckerSymbol |