# Pépin’s theorem

Theorem (Pépin). A Fermat number ${F}_{n}={2}^{{2}^{n}}+1$ is prime only if

$${3}^{\frac{{F}_{n}-1}{2}}\equiv -1mod{F}_{n}.$$ |

In other words, if 3 raised to the largest power of two not greater than the Fermat number leaves as a remainder the next higher power of two when divided by that Fermat number (since ${F}_{n}-1={2}^{{2}^{n}}$), then that Fermat number is a Fermat prime.

For example, $17={2}^{{2}^{2}}+1$ is a Fermat prime, and we can see that ${3}^{8}=6561$, which leaves a remainder of 16 when divided by 17. The smallest Fermat number not to be a prime is 4294967297, as it is the product of 641 and 6700417, and ${3}^{2147483648}$ divided by 4294967297 leaves a remainder of 10324303 rather than 4294967296.

Title | Pépin’s theorem |
---|---|

Canonical name | PepinsTheorem |

Date of creation | 2013-03-22 18:53:09 |

Last modified on | 2013-03-22 18:53:09 |

Owner | PrimeFan (13766) |

Last modified by | PrimeFan (13766) |

Numerical id | 4 |

Author | PrimeFan (13766) |

Entry type | Theorem |

Classification | msc 11A51 |

Synonym | Pepin’s theorem |