Definition:Fermat Prime

Definition
A Fermat prime is a Fermat number, i.e. a number of the form $2^{\left({2^n}\right)} + 1$, which happens to be prime.

In fact, $2^{\left({2^n}\right)} + 1$ is prime for $n = 0, 1, 2, 3, 4$.

However, $2^{\left({2^5}\right)} + 1 = 2^{32} + 1$ is divisible by $641$, as was proved by Euler.

No Fermat primes for $n > 4$ have ever been discovered.

Examples
The only known examples of Fermat primes are as follows:

He (incorrectly) conjectured that all numbers of the form $2^{\left({2^n}\right)} + 1$ are prime.