Are there any more Fermat Primes?

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Open Question

Are there any more Fermat primes than the $5$ that are known about?


Sequence

The sequence of Fermat primes begins:

\(\displaystyle 2^{\paren {2^0} } + 1\) \(=\) \(\displaystyle 3\)
\(\displaystyle 2^{\paren {2^1} } + 1\) \(=\) \(\displaystyle 5\)
\(\displaystyle 2^{\paren {2^2} } + 1\) \(=\) \(\displaystyle 17\)
\(\displaystyle 2^{\paren {2^3} } + 1\) \(=\) \(\displaystyle 257\)
\(\displaystyle 2^{\paren {2^4} } + 1\) \(=\) \(\displaystyle 65 \, 537\)

No other Fermat primes are known.


Sources