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.