The Existence of Infinitely Many Fermat Primes

Fermat numbers are \(F_{k}\) = \(2^{2^{k}}\) + 1. \(F_{0}\) through \(F_{4}\) are prime (3, 5, 17, 257, 65537). Question: are there infinitely many Fermat primes? Or equivalently: does \(F_{k}\) produce a prime for any k > 4?

Fermat conjectured in 1640 that ALL Fermat numbers are prime. Euler demolished this in 1732 by finding 641 | \(F_{5}\). Since then, every \(F_{5}\) through \(F_{32}\) has been shown composite (some requiring massive distributed computation). Heuristics suggest only finitely many exist (expected total: ~5-6 based on probabilistic models), yet no proof rules out more. Each new candidate requires testing numbers with millions of digits.