A Fermat Number is a positive integer of the form $F(n) = 2$ to the ${2}^{n}$ power + 1. It was discovered and named after Pierre de Fermat.

The first two Ferman Numbers are as follows:

$F(0) = 2^{1} + 1 = 3$

$F(1) = 2^{2} + 1 = 5$

As of 2008, only the first eleven Fermat Numbers has been factored.

In regards to primes, a Fermat Prime is any prime of the form $2^{n} + 1$. The only Fermat Numbers that are Fermat primes, as of right now, are the first 5, F(0) to F(4).

One issue relating to Fermat Numbers and Primes is that Fermat Numbers with large n's have not been studied very much. As of right now, there are only 5 Fermat Primes, but their may be more. Because of this, three questions persist:

Is Fn composite for all n > 4?

Are there infinitely many Fermat primes?

Are there infinitely many composite Fermat numbers?

As of 2008 it is known that Fn is composite for 5 ≤ n ≤ 32, and the largest composite Fermat Number is F(2478782), which has the prime factorization of $3 x 2^{2478785} + 1.$

Fermat Primes have become a more studied topic in number theory in recent years, and as the course goes on, we may learn more about mathematical properties that relate to these primes. Modular arithmetic is used in congruence with Fermat Primes for some proofs, so we will better be able to understand Fermat Primes as the course goes on.