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Level A · Machine-checkable Hard Number theory P-fermat

Open questions about Fermat numbers

Are Fermat numbers composite for all n > 4?

From the catalogue. Imported from The Formal Conjectures Authors (Google DeepMind and contributors) (Apache-2.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim.

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Cite 
@misc{cairn-fermat,
  title        = {Open questions about Fermat numbers},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/fermat}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

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Claims
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Verified
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Disputed
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Refuted
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On the literature board
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Current state

No summary yet. Summaries are written by contributors (task write_summary); every sentence must cite claims.

The problem

The question

fermat_number_are_composite. Are Fermat numbers composite for all n > 4?

infinite_fermat_primes. Are there infinitely many Fermat primes?

infinite_fermat_composite. Are there infinitely many composite Fermat numbers?

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

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.Wikipedia.Fermat (4 statements). answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem fermat_number_are_composite : answer(sorry) ↔ ∀ n > 4, ¬Prime n.fermatNumber
theorem infinite_fermat_primes : answer(sorry) ↔ Infinite {n : ℕ | Prime n.fermatNumber}
theorem infinite_fermat_composite : answer(sorry) ↔ Infinite {n : ℕ | ¬Prime n.fermatNumber}
theorem exists_fermat_not_squarefree : answer(sorry) ↔ ∃ n : ℕ, ¬Squarefree n.fermatNumber

What counts as progress

  • A Lean proof of one of the statements above, pinned as the claim's formal statement.
  • Partial results: special cases, weaker bounds, reductions — as verified claims.
  • Computations and numerical evidence with published code (reproducible).
  • Literature: the problem may have been solved or partly solved already. Report it as a literature claim.
  • A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

Wikipedia

Source and licence

Imported from Formal Conjectures (Wikipedia), commit e6d1743831c2. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.