Constructing regular polygons Compass-and-straightedge construction

construction of regular pentagon

some regular polygons (e.g. pentagon) easy construct straightedge , compass; others not. led question: possible construct regular polygons straightedge , compass?

carl friedrich gauss in 1796 showed regular 17-sided polygon can constructed, , 5 years later showed regular n-sided polygon can constructed straightedge , compass if odd prime factors of n distinct fermat primes. gauss conjectured condition necessary, offered no proof of fact, provided pierre wantzel in 1837.

the first few constructible regular polygons have following numbers of sides:

3, 4, 5, 6, 8, 10, 12, 15, 16, 17, 20, 24, 30, 32, 34, 40, 48, 51, 60, 64, 68, 80, 85, 96, 102, 120, 128, 136, 160, 170, 192, 204, 240, 255, 256, 257, 272... (sequence a003401 in oeis)

there known infinitude of constructible regular polygons number of sides (because if regular n-gon constructible, regular 2n-gon , hence regular 4n-gon, 8n-gon, etc.). however, there 31 known constructible regular n-gons odd number of sides.