The current status of the Fermat numbers $$F_n=2^{2^n}+1$$ where $n$ is a nonnegative integer , is that it is prime for $n\le 4$ and composite for $5\le n\le 32$
It is conjectured that only finitely many Fermat primes exist , the probability of a further Fermat prime is less than $1:10^9$ and the next possible is $F_{33}$.
It seems therefore absurd that $F_n$ is prime for every sufficiently large $n$ , in other words , that only finite many Fermat numbers are composite. But such proofs are usually out of reach. Is this actually an open problem ? Is it true that we cannot rule out this scenario ?