Fermat.lean`

Explanation legend Lean4 Domain Explanation Language (L4DEL)

L4DEL explanations are a separate layer. They do not change the Lean source. Intuition is explanatory prose; numbered items are a reading path; sources link the wording back to evidence.

L4DEL: Lean4 Domain Explanation Language

Annotation provenance. This overlay contains 10 curated AI-written annotations and 7190 mechanical source-derived annotations. They have not been human-reviewed as explanations; treat them as unverified and not mathematical authority. Retained generation history: The 71 original curated annotations were produced in a Claude Sonnet 5.5 session at medium effort; Claude Opus 5.5 is recorded in that session as the advisor model, not the direct author. The bulk source-derived expansion was generated mechanically from pinned source comments by tooling developed in a GPT-5.6 Sol session at ultra reasoning; GPT-5.6 Sol did not originate the quoted source prose. On 2026-10-17 the mechanically generated source-comment annotations of the Formal Conjectures, Carleson, FLT, PFR, CompFiles, Mathematics in Lean and Theorem Proving in Lean overlays were regenerated by the same generator after three corrections made in a Claude Sonnet 5.5 session: titles no longer cut a sentence inside mathematics or end on a dangling word, workflow markers are removed from the quoted prose, and declaration names follow the scanner's namespace resolution. The boilerplate step line was dropped, and comments that were only a row of dashes no longer yield annotations. No new prose was written and no human review is claimed.

Credit: The upstream authors and contributors retain credit for the project and mathematics. lean4-explainer only renders their pinned source and adds the separately disclosed explanation layer.

Project sources: Pinned project documentation, Formal Conjectures paper, arXiv v1

4 declarations, 4 contain sorry, 0 declared as axioms

Open questions about Fermat numbers

Reference: Wikipedia

Imports (1)

FormalConjecturesUtil

theorem Fermat.fermat_number_are_composite Are Fermat numbers composite for all n > 4? sorry line 34 source
Source documentation
Are Fermat numbers composite for all n > 4?

depends on sorry what this means
Statement uses: Iff, True, Nat, GT.gt, OfNat.ofNat, Not, Prime, Nat.fermatNumber
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
6 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem fermat_number_are_composite : answer(sorry) ↔ ∀ n > 4, ¬Prime n.fermatNumber := by
…
```
theorem Fermat.infinite_fermat_primes Are there infinitely many Fermat primes? sorry line 41 source
Source documentation
Are there infinitely many Fermat primes?

depends on sorry what this means
Statement uses: Iff, True, Infinite, Nat, Set.ofPred, Prime, Nat.fermatNumber
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
5 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem infinite_fermat_primes : answer(sorry) ↔ Infinite {n : ℕ | Prime n.fermatNumber} := by
…
```
theorem Fermat.infinite_fermat_composite Are there infinitely many composite Fermat numbers? sorry line 48 source
Source documentation
Are there infinitely many composite Fermat numbers?

depends on sorry what this means
Statement uses: Iff, True, Infinite, Nat, Set.ofPred, Not, Prime, Nat.fermatNumber
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
5 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem infinite_fermat_composite : answer(sorry) ↔ Infinite {n : ℕ | ¬Prime n.fermatNumber} := by
…
```
theorem Fermat.exists_fermat_not_squarefree Does a Fermat number exist that is not square-free? sorry line 55 source
Source documentation
Does a Fermat number exist that is not square-free?

depends on sorry what this means
Statement uses: Iff, True, Exists, Nat, Not, Squarefree, Nat.fermatNumber
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
3 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem exists_fermat_not_squarefree : answer(sorry) ↔ ∃ n : ℕ, ¬Squarefree n.fermatNumber := by
…
```