Fermat numbers

Any number in the form $2^{2^n}+1$ where $n$ is any natural number is known as a Fermat number. It was hypothesized by Fermat that every number in this form was prime, but Euler found that the fifth Fermat number can be factored as $2^{2^5}+1=641 \cdot 6,700,417$. There are only five known Fermat Primes, and it is believed that there are only five, but we are still lacking a complete proof.

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