A number of the form for nonnegative integer is a Fermat number. The first five Fermat numbers (for ) are and each of these is a Fermat prime. 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.
Primes one more than a power of 2
Fermat primes are also the only primes in the form .
Suppose that has an odd factor . For all odd , we have by the Root-Factor Theorem that divides . Since this is true as a statement about polynomials, it is true for every integer value of . In particular, setting gives that , and since , this shows that is not prime.
It follows that if is prime, must have no odd factors other than and so must be a power of 2.