Editing Constructible polygon (section)

===Detailed results by Gauss's theory===

Restating the Gauss–Wantzel theorem:
:A regular ''n''-gon is constructible with straightedge and compass if and only if ''n'' = 2<sup>''k''</sup>''p''<sub>1</sub>''p''<sub>2</sub>...''p''<sub>''t''</sub> where ''k'' and ''t'' are non-negative [[integer]]s, and the ''p''<sub>''i''</sub>'s (when ''t'' > 0) are distinct Fermat primes.

The five known [[Fermat primes]] are:
:''F''<sub>0</sub> = 3, ''F''<sub>1</sub> = 5, ''F''<sub>2</sub> = 17, ''F''<sub>3</sub> = 257, and ''F''<sub>4</sub> = 65537 {{OEIS|id=A019434}}.

Since there are 31 nonempty subsets of the five known Fermat primes, there are 31 known constructible polygons with an odd number of sides.

The next twenty-eight Fermat numbers, ''F''<sub>5</sub> through ''F''<sub>32</sub>, are known to be [[composite number|composite]].<ref>[http://www.prothsearch.com/fermat.html Prime factors  k · 2n + 1  of Fermat numbers  Fm and complete factoring status] by Wilfrid Keller.</ref>

Thus a regular ''n''-gon is constructible if
:''n'' = [[Equilateral triangle|3]], [[Square|4]], [[Pentagon|5]], [[Hexagon|6]], [[Octagon|8]], [[Decagon|10]], [[Dodecagon|12]], [[Pentadecagon|15]], [[Hexadecagon|16]], [[Heptadecagon|17]], [[Icosagon|20]], [[Icositetragon|24]], [[Triacontagon|30]], 32, 34, 40, 48, 51, 60, 64, 68, 80, 85, 96, 102, 120, 128, 136, 160, 170, 192, 204, 240, 255, 256, [[257-gon|257]], 272, 320, 340, 384, 408, 480, 510, 512, 514, 544, 640, 680, 768, 771, 816, 960, 1020, 1024, 1028, 1088, 1280, 1285, 1360, 1536, 1542, 1632, 1920, 2040, 2048, ... {{OEIS|id=A003401}},

while a regular ''n''-gon is not constructible with compass and straightedge if
:''n'' = [[Heptagon|7]], [[Enneagon|9]], [[Hendecagon|11]], [[Tridecagon|13]], [[Tetradecagon|14]], [[Octadecagon|18]], 19, 21, 22, [[Icositrigon|23]], 25, 26, 27, 28, 29, 31, 33, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 49, 50, 52, 53, 54, 55, 56, 57, 58, 59, 61, 62, 63, 65, 66, 67, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 81, 82, 83, 84, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 97, 98, 99, 100, 101, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 121, 122, 123, 124, 125, 126, 127, ... {{OEIS|id=A004169}}.