For those curious…
If you can make a 3-gon and a 5-gon, and you do so in the same circle, you can find the length needed to make a 15-gon. With that sort of trick one can reduce to p^n-gons where p is prime.
Gauss figured out that you can make a p-gon iff p is 1 more than a power of 2. This is basically because one’s trying to find the complex numbers solving z^p=1, but not z=1, so one faces a polynomial of degree p-1. Why power of 2? Because intersecting a line with a circle, or circle with circle, lets you find square roots of numbers you already have. Then fourth roots, eighth roots, …
I had a homework problem once to invent the construction, but not to actually carry it out. Supposedly there’s a satchel in Gottingen full of the details of the 65537-gon.
