Fermat numbers

The n-th Fermat number is defined as:


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 divisorMathworldPlanetmathPlanetmath of F5. (In fact, F5=641×6700417). 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 polygonMathworldPlanetmath of m sides can be constructed with ruler and compass if and only if m can be written as


where k0 and the other factors are distinct primes of the form Fn (of course, t may be 0 here, i.e. m=2k is allowed).

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


for any m1, which implies that Fm-2 is divisible by all smaller Fermat numbers.

The previous formula holds because


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

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)
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Defines Fermat prime