# Fermat prime

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 .