Fermat prime

A Fermat prime, named after Pierre de Fermat who first studied them, is a prime number of the form:

<math> F_{n} = 2^{2^n} + 1 </math>

where n is a natural number. There are only five known Fermat primes: 3 (n=0), 5 (n=1), 17 (n=2), 257 (n=3) and 65537 (n=4). It is not known whether these are the only Fermat primes, and it is not even known whether or not there are infinitely many Fermat primes.

Carl Friedrich Gauss proved that there is a relationship between the ruler and compass construction of regular polygons and Fermat primes: a regular n-gon can be constructed with ruler and compasses if and only if n is a power of 2 or the product of a power of 2 and distinct Fermat primes.

Integers of the general form

<math>2^{2^n}+1 </math>

with n a natural number are known as Fermat numbers. Fermat conjectured that all of them were prime numbers, but was later proven wrong when the Fermat number for n=5 was shown to be composite by Leonhard Euler in 1732. We have:

<math> F_{5} = 2^{2^5} + 1 = 2^{32} + 1 = 4294967297 = 641 \cdot 6700417 \; </math>

Different Fermat numbers are relatively prime.

See also:

• Mersenne prime -- Gaussian prime

External links:

• Generalized Fermat Prime search (http://perso.wanadoo.fr/yves.gallot/primes/gfn.html)
• http://primes.utm.edu/glossary/page.php?sort=Fermats