FormalConjectures.Wikipedia.Fermat

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import FormalConjecturesUtil

Open questions about Fermat numbers

Reference: Wikipedia


namespace Fermat

Are Fermat numbers composite for all n > 4?

@[category research open, AMS 11]
theorem declaration uses 'sorry'fermat_number_are_composite : answer(sorry) ↔ ∀ n > 4, ¬Prime n.fermatNumber := by⊢ True ↔ ∀ n > 4, ¬Prime n.fermatNumber
  sorryAll goals completed! 🐙

Are there infinitely many Fermat primes?

@[category research open, AMS 11]
theorem declaration uses 'sorry'infinite_fermat_primes : answer(sorry) ↔ Infinite {n : ℕ | Prime n.fermatNumber} := by⊢ True ↔ Infinite ↑{n | Prime n.fermatNumber}
  sorryAll goals completed! 🐙

Are there infinitely many composite Fermat numbers?

@[category research open, AMS 11]
theorem declaration uses 'sorry'infinite_fermat_composite : answer(sorry) ↔ Infinite {n : ℕ | ¬Prime n.fermatNumber} := by⊢ True ↔ Infinite ↑{n | ¬Prime n.fermatNumber}
  sorryAll goals completed! 🐙

Are all Fermat numbers are square-free?

@[category research open, AMS 11]
theorem declaration uses 'sorry'all_fermat_squarefree : answer(sorry) ↔ ∀ n : ℕ, Squarefree n.fermatNumber := by⊢ True ↔ ∀ (n : ℕ), Squarefree n.fermatNumber
  sorryAll goals completed! 🐙


end Fermat