Fermat Prime 

A regular polygon with n sides is constructible with ruler and compass if and only if n is the product of 0 or more distinct Fermat primes, and a power of 2.
Exercise: prove that if $2^m+1$ is prime, then m is a power of 2. (Hint: if m is not a power of 2 then it has an odd factor other than 1. Can you use this to find a factor of $2^m+1$ ?)
Another exercise: prove that $2^32+1$ is a multiple of 641 without doing any nontrivial arithmetic, using the fact that $641=5.2^7+1=5^4+2^4.$
See also: Mersenne prime.