Definition:Fermat Prime

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.

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

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