# Fermat numbers

The $n$-th Fermat number is defined as:

 $F_{n}=2^{2^{n}}+1.$

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

 $m=2^{k}F_{r_{1}}F_{r_{2}}\cdots F_{r_{t}}$

where $k\geq 0$ 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:

 $F_{m}=F_{0}F_{1}\cdots F_{m-1}+2$

for any $m\geq 1$, 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