# Agoh-Giuga conjecture

In 1950, Giuseppe Giuga conjectured that if and only if an integer $p$ is prime then it will satisfy the congruence^{}

$$\sum _{i=1}^{p-1}{i}^{p-1}\equiv -1modp.$$ |

This is sometimes called the *Giuga conjecture ^{}*. Takashi Agoh rephrased the conjecture as $p{B}_{p-1}\equiv -1modp$, where $B$ is a Bernoulli number

^{}; this is called the

*Agoh-Giuga conjecture*. In 2003 Simon Plouffe performed an exhaustive search for a counterexample below 50000 but came up empty.

Title | Agoh-Giuga conjecture |
---|---|

Canonical name | AgohGiugaConjecture |

Date of creation | 2013-03-22 16:17:55 |

Last modified on | 2013-03-22 16:17:55 |

Owner | PrimeFan (13766) |

Last modified by | PrimeFan (13766) |

Numerical id | 6 |

Author | PrimeFan (13766) |

Entry type | Conjecture |

Classification | msc 11D85 |

Synonym | Giuga conjecture |

Synonym | Giuga’s conjecture |

Synonym | Agoh conjecture |

Synonym | Agoh’s conjecture |