Existence of additional Fermat primes

Determine whether there exist any other Fermat primes of the form 2^{2^k}+1 beyond the currently known set {3, 5, 17, 257, 65537}, in order to complete the characterization of regular n-gon constructibility by ruler and compass, which depends on the prime factorization of n into powers of 2 and distinct Fermat primes.

Background

A Fermat prime is a prime of the form 2^{2^k}+1. In the classical ruler-and-compass problem, a regular n-gon is constructible if and only if n is of the form 2^r p_1 ··* p_k where the p_i are distinct Fermat primes. Thus, the set of Fermat primes directly determines which regular polygons are constructible.

Only five Fermat primes are currently known: 3, 5, 17, 257, and 65537. The status of any further primes of this special form remains unresolved, and settling their existence would have immediate consequences for the complete classification of constructible regular polygons.