If is a nonnegative integer the Fermat number is defined to be .
If is prime, then it is known as a Fermat prime. The first Fermat numbers (for ) are known to be prime. Indeed: Based on these results, one might conjecture (as did Fermat) that all Fermat numbers are prime. However, this fails for : . In fact the primes listed above are the only Fermat numbers known to be prime.
Fermat primes are also the only primes in the form . This is easy to see, if has an odd factor then , a contradiction, hence is a power of .