Euclid.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

3 declarations, 2 contain sorry, 0 declared as axioms

Euclid Numbers conjecture

Reference: Wikipedia

Imports (1)

FormalConjecturesUtil

def EuclidNumbers.Euclid The n-th Euclid number is the product of the first n prime numbers plus one line 33 source
Source documentation
The n-th Euclid number is the product of the first n prime numbers plus one.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: HAdd.hAdd, OfNat.ofNat, Finset.prod, Finset.range, Nat.nth, Nat.Prime
4 instances, coercions and auxiliary constants left out.
```
noncomputable def Euclid (n : ℕ) : ℕ := 1 + ∏ i ∈ Finset.range n, i.nth Nat.Prime
…
```
theorem EuclidNumbers.infinite_prime_euclid_numbers It is not known whether there is an infinite number of prime Euclid numbers sorry line 39 source
Source documentation
It is not known whether there is an infinite number of prime Euclid numbers.

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

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem infinite_prime_euclid_numbers : answer(sorry) ↔ {n | (Euclid n).Prime}.Infinite := by
…
```
Mentions: EuclidNumbers.Euclid
theorem EuclidNumbers.euclid_numbers_are_square_free It is not known whether every Euclid number is a square-free number sorry line 46 source
Source documentation
It is not known whether every Euclid number is a square-free number.

depends on sorry what this means
Statement uses: Iff, True, Nat, Squarefree, EuclidNumbers.Euclid
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 euclid_numbers_are_square_free : answer(sorry) ↔ (∀ n, Squarefree (Euclid n)) := by
…
```
Mentions: EuclidNumbers.Euclid