Comment by plg94

For 17, Gauss noticed that cos(360°/17) can be written only with elementary operations, see https://www.heise.de/imgs/18/2/1/2/3/3/6/4/siebzehneck-b95b5...

Later he proved that all n-gons with $n=2^k*p_1…*p_r$ where the p_i are Fermat-primes (2^(2^m)+1 prime, today we only know of 3, 5, 17, 257, 65537) are constructible. The opposite direction, i.e. all other n are not constructible, was only a few years later proved. Look up "Theorem of Gauss-Wantzel". I only skimmed the proof, but it seems to generalize the concept of constructing the cos of the angle with "Galois-Theory".

(edit: or see https://en.wikipedia.org/wiki/Constructible_polygon)