# Fermat numbers are coprime

###### Theorem.

Any two Fermat numbers are coprime.

Proof.

Let ${F}_{m}$ and ${F}_{n}$ two Fermat numbers, and assume $$.
Let $d$ a positive^{} common divisor^{} of ${F}_{n}$ and ${F}_{m}$, that is

$$d\mid {F}_{m},d\mid {F}_{n}.$$ |

If $d\mid {F}_{m}$ then $d\mid {F}_{1}{F}_{2}\mathrm{\cdots}{F}_{n-1}$ since some factor must be ${F}_{m}$ itself. But ${F}_{n}-{F}_{1}{F}_{2}\mathrm{\cdots}{F}_{n-1}=2$ and so $d\mid 2$. Since $d$ is odd, we must have $d=1$.

Therefore, the greatest common divisor^{} of any two Fermat numbers must be $1$.

Q.E.D.

Title | Fermat numbers are coprime |
---|---|

Canonical name | FermatNumbersAreCoprime |

Date of creation | 2013-03-22 14:51:24 |

Last modified on | 2013-03-22 14:51:24 |

Owner | yark (2760) |

Last modified by | yark (2760) |

Numerical id | 5 |

Author | yark (2760) |

Entry type | Theorem |

Classification | msc 11A51 |