# Fermat numbers

Fermat incorrectly conjectured that all these numbers were primes,
although he had no proof.
The first 5 Fermat numbers: $3,5,17,257,65537$ (corresponding to $n=0,1,2,3,4$) are all primes (so called Fermat primes)
Euler was the first to point out the falsity of Fermat’s conjecture
by proving that $641$ is a divisor^{} of ${F}_{5}$. (In fact, ${F}_{5}=641\times 6700417$).
Moreover, no other Fermat number is known to be prime for $n>4$, so now it is conjectured that those are all prime Fermat numbers. It is also unknown whether there are infinitely many composite Fermat numbers or not.

One of the famous achievements of Gauss was to prove that the regular polygon^{} of $m$ sides can be constructed with ruler and compass if and only if $m$ can be written as

$$m={2}^{k}{F}_{{r}_{1}}{F}_{{r}_{2}}\mathrm{\cdots}{F}_{{r}_{t}}$$ |

where $k\ge 0$ and the other factors are distinct primes of the form ${F}_{n}$ (of course, $t$ may be $0$ here, i.e. $m={2}^{k}$ is allowed).

There are many interesting properties involving Fermat numbers. For instance:

$${F}_{m}={F}_{0}{F}_{1}\mathrm{\cdots}{F}_{m-1}+2$$ |

for any $m\ge 1$, which implies that ${F}_{m}-2$ is divisible by all smaller Fermat numbers.

The previous formula holds because

$${F}_{m}-2=({2}^{{2}^{m}}+1)-2={2}^{{2}^{m}}-1=({2}^{{2}^{m-1}}-1)({2}^{{2}^{m-1}}+1)=({2}^{{2}^{m-1}}-1){F}_{m-1}$$ |

and expanding recursively the left factor in the last expression gives the desired result.

References.

Krízek, Luca, Somer. *17 Lectures on Fermat Numbers.* CMS Books in Mathematics.

Title | Fermat numbers |

Canonical name | FermatNumbers |

Date of creation | 2013-03-22 11:42:46 |

Last modified on | 2013-03-22 11:42:46 |

Owner | drini (3) |

Last modified by | drini (3) |

Numerical id | 30 |

Author | drini (3) |

Entry type | Definition |

Classification | msc 81T45 |

Classification | msc 81T13 |

Classification | msc 11A51 |

Classification | msc 20L05 |

Classification | msc 46L87 |

Classification | msc 43A35 |

Classification | msc 43A25 |

Classification | msc 22D25 |

Classification | msc 55U40 |

Classification | msc 18B40 |

Classification | msc 46L05 |

Classification | msc 22A22 |

Classification | msc 81R50 |

Classification | msc 55U35 |

Defines | Fermat prime |