Fermat prime
A Fermat prime is a prime number of the form for some nonnegative integer
.
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 .
Proof
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.