In mathematics, a Fermat number, named after Pierre de Fermat who first studied them, is an integer of the form $$F_{n} = 2^{2^n} + 1$$ where $n$ is a nonnegative integer. The first few Fermat numbers are: $$3,\ 5,\ 17,\ 257,\ 65537,\ \cdots $$
If $2^k +1$ is prime, and $k > 0$, it can be shown that $k$ must be a power of two, $k=2^n$. A number of the form $2^{2^n}+1$ is called a Fermat number, and when it happens to be a prime, it is called a Fermat prime. As of 2017, the only known Fermat primes are those for which $0\leq n\leq 4$. In addition, John L. Selfridge made an intriguing conjecture: Let $g(n)$ be the number of distinct prime factors of $2^{2^n} + 1$. Then $g(n)$ is not monotonic (non-decreasing).
If another Fermat prime exists, that would imply that the conjecture is false.
