FormalConjectures.Wikipedia.Fermat

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you may not use this file except in compliance with the License.
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module

public import FormalConjecturesUtil

Open questions about Fermat numbers

Reference: Wikipedia

@[expose] public sectionnamespace Fermat

Are Fermat numbers composite for all n > 4?

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

Does a Fermat number exist that is not square-free?

@[category research open, AMS 11]
theorem exists_fermat_not_squarefree : answer(sorry) ↔ ∃ n : ℕ, ¬Squarefree n.fermatNumber := by⊢ True ↔ ∃ n, ¬Squarefree n.fermatNumber
  sorryAll goals completed! 🐙

end Fermat