Fermat numbers

Any number in the form $2^{2^n}+1$ where $n$ is any natural number is known as a Fermat number. It was hypothesized by Fermat that every number in this form was prime, but Euler found that the fifth Fermat number can be factored as $2^{2^5}+1=641 \cdot 6,700,417$ . There are only five known Fermat Primes, and it is believed that there are only five, but we are still lacking a complete proof.