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  "body": "\n## Definition\n\nA Fermat prime is a [[Fermat number|0R03B54V4Z]] $F_n = 2^{2^n} + 1$ that is a [[prime number|0SN0MAEKYX]].\n\n## Remarks\n\nThe numbers $F_0, \\dots, F_4$, that is $3, 5, 17, 257, 65537$, are prime. No other Fermat prime is known.\nFermat conjectured that all Fermat numbers are prime. Euler refuted this in 1732 with $F_5 = 641 \\cdot 6700417$.\nA regular $n$-gon is constructible with ruler and compass if and only if $n$ is a power of $2$ times distinct Fermat primes.\nThis is the Gauss–Wantzel theorem.\n\n## Explanations\n\n### L1 human\n\nA Fermat prime is a prime of the form $2^{2^n} + 1$.\nOnly five of them are known, and it is an open problem whether there are more.\n",
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