Drawing regular polygons

Which regular polygons can you draw with a pencil, compass, and straightedge? Is there some formula to it? At first, it might appear to be a bit random. 3,4,5,6, and 8 sided shapes all can be drawn. However, there is a rule for this.

Any regular shape that has a product of a power of 2. can be drawn This makes sense, because you can keep on splitting a square or hexagon into smaller and smaller pieces, but you can't split a heptagon into smaller pieces because you can't construct it in the first place.

For odd numbers though, the of regular sides have to be a Fermat prime, which is defined as 2^(2^n) + 1. When we substitute n for the first few positive integers, (plus zero) we get:

2^(2^0) + 1 = 2^1 + 1 = 3

2^(2^1) + 1 = 2^2 + 1 = 4 + 1 = 5

2 ^(2^2) + 1 = 2^4 + 1 = 16 + 1 = 17

All regular polygons with those number of sides can be constructed.

This sequence gets inflated quite quickly, as here are the next few terms: 257, 65537, 4294967297, and 18446744073709551617.

So to anyone who wants to construct a 18446744073709551617-gon, I wish you good luck as that might take you a very long while.