Open questions about Fermat numbers

Reference: Wikipedia

theorem Fermat.fermat_number_are_composite : (∀ n > 4, ¬Prime n.fermatNumber) ↔ sorry

Are Fermat numbers composite for all n > 4?

theorem Fermat.infinite_fermat_primes : Infinite ↑{n : ℕ | Prime n.fermatNumber} ↔ sorry

Are there infinitely many Fermat primes?

theorem Fermat.infinite_fermat_composite : Infinite ↑{n : ℕ | ¬Prime n.fermatNumber} ↔ sorry

Are there infinitely many composite Fermat numbers?

theorem Fermat.all_fermat_squarefree : (∀ (n : ℕ), Squarefree n.fermatNumber) ↔ sorry

Are all Fermat numbers are square-free?