[R. Gerbicz]

Originally Posted by Dr Sardonicus

You might like Constructing 17, 257, and 65537 sided polygons

See also the references in Wolfram Mathworld's 257-gon. The one by author Richelot, F. J. looks like it's right up your alley.

Still in high school's math camp we constructed a regular 17-gon.

Originally Posted by a1call

Similarly for 3*17=51 gon
1/3-3/17= 8/51 of a circle. Then you can bisect 3 times to get 1/51 of a circle.

There is an elementary way to show that if you can make a regular m and n-gon [using straightedge and compass] and gcd(m,n)=1 then you can make a regular m*n-gon. Because making a regular k-gon is equivalent with constructing a 2*Pi/k angle.

So we can make a 2*Pi/n and 2*Pi/m angle.

We assumed that gcd(m,n)=1 so with extended Euclidean algorithm there exists x and y integers:

n*x+m*y=1 divide this equation by m*n

x/m+y/n=1/(m*n) multiplie by 2*Pi

x*(2*Pi/m)+y*(2*Pi/n)=2*Pi/(m*n), what we needed.