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  "body": "\n## Statement\n\nLet $n \\ge 3$.\nThe regular $n$-gon can be constructed with straightedge and compass if and only if $n = 2^k p_1 p_2 \\cdots p_t$, with $k, t \\ge 0$ and $p_1, \\dots, p_t$ distinct Fermat primes.\n\n## Remarks\n\nA Fermat prime is a prime number of the form $2^{2^m} + 1$. The known Fermat primes are $3, 5, 17, 257$ and $65537$.\nThe regular $n$-gon is constructible if and only if $e^{2\\pi i/n}$ is a [[constructible number|0FMN2MN7TM]]. Its degree over $\\mathbb{Q}$ is $\\varphi(n)$, by the [[irreducibility of $\\Phi_n$|0SK1VHZ5R2]].\nThe condition on $n$ holds if and only if $\\varphi(n)$ is a power of $2$.\nSo the $17$-gon and the $15$-gon are constructible, but the $7$-gon and the $9$-gon are not.\n\n## History\n\nCarl Friedrich Gauss proved in 1796 that the regular $17$-gon is constructible.\nIn his 1801 book Disquisitiones Arithmeticae, he gave the sufficient condition.\nPierre Wantzel gave a full proof of the necessity in 1837.\n\n## Explanations\n\n### L1 human\n\nThe theorem tells exactly which regular polygons you can draw with straightedge and compass.\nThe answer depends on the prime factors of the number of sides.\nOnly Fermat primes, each used at most once, and powers of $2$ are allowed.\n",
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