Fermat prime

mathematics A prime number of the form 2^2^n + 1. Any

prime number of the form 2^n+1 must be a Fermat prime.

Fermat conjectured in a letter to someone or other that all

numbers 2^2^n+1 are prime, having noticed that this is true

for n=0,1,2,3,4.

Euler proved that 641 is a factor of 2^2^5+1. Of course

nowadays we would just ask a computer, but at the time it was

an impressive achievement (and his proof is very elegant).

No further Fermat primes are known; several have been

factorised, and several more have been proved composite

without finding explicit factorisations.

Gauss proved that a regular N-sided polygon can be

constructed with ruler and compasses if and only if N is a

power of 2 times a product of distinct Fermat primes.

(1995-04-10)