Open questions about Fermat numbers
Are Fermat numbers composite for all n > 4?
Current state (tracked automatically)
All tracked facts and sources
| Fact | Value | Source |
|---|---|---|
| Formal Conjectures: research statements in the file | 4 open, 0 solved | Formal Conjectures |
Read twice a day from public sources by deterministic collectors (no AI; the sentences above are filled-in templates): Formal Conjectures (checked 2026-11-12). How it works.
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.
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 2026-11-12}
} Also: CITATION.cff · Atom feed of results
Status badge for a README (shields.io):
[](https://cairn-commons.com/problems/fermat)
- Claims
- 0
- Verified
- 0
- Disputed
- 0
- Refuted
- 0
- On the literature board
- 0
Nobody has worked on this problem here yet
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What a result looks like: C-16, a newcomer working with an AI model found C(85,24,3) ≤ 76 (previous best 77), and a computer verified it within minutes. A documented dead end counts too.
ChatGPT opens in a new tab and starts on the task right away; Claude opens with it filled in.
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).
theorem fermat_number_are_composite : ∀ n > 4, ¬Prime n.fermatNumbertheorem infinite_fermat_primes : Infinite {n : ℕ | Prime n.fermatNumber}theorem infinite_fermat_composite : Infinite {n : ℕ | ¬Prime n.fermatNumber}theorem exists_fermat_not_squarefree : ∃ n : ℕ, ¬Squarefree n.fermatNumberWhat 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
Source and licence
Imported from Formal Conjectures (Wikipedia), commit 2f91d251c07f. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.