Constructible polygons are polygons that can be constructed using only straight lines and circles. The rules for constructing these polygons are that you can only use straight lines and circles, obviously, by using a ruler and a compass. However, one is not allowed to make use of any other properties of the ruler and compass, such as measuring.
Gauss' Conjecture in Relation to Constructible Polygons
Gauss proposed that a regular polygon can be constructed using only straight lines if its odd prime factors are distinct Fermat numbers (i.e. numbers which follow ). This means that 2 is the only prime factor that doesn't count. Thus, even numbers whose odd prime factors are Fermat numbers are constructible. There are composite Fermat numbers which begin at when n is 5. This was later proven by Pierre Wantzel. The known prime Fermat numbers are as follows: 3, 5, 17, 257 and 65537. Thus, the first few n-gons that can be constructed are: 3, 4, 5, 6, 8, 10, 12, 15 and 16; the odd prime Fermat number factors being 3, none, 5, 3, none, 5, 3, 3 and 5, none.
Although Gauss proved that heptadecagons (17-gons) as well as 257-gons and 65537-gons are constructible polygons, he did not show how to construct any of them, and it was first shown by Erchinger. The first demonstrations of a 257-gon (the next Fermat prime) was given by Magnus Paucker. A demonstration for a 65537-gon was first given by Johann Hermes.