Lemoine.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

Lemoine's conjectures

References:

Wikipedia
[Ki85] Kiltinen, J. and Young P. (1985). Goldbach, Lemoine, and a Know/Don't Know Problem.

Imports (1)

FormalConjecturesUtil

def Lemoine.OddPrime line 32 source
standard axioms: propext what this means
Statement uses: Nat
Proof uses: And, Ne, OfNat.ofNat, Nat.Prime
1 instances, coercions and auxiliary constants left out.
theorem Lemoine.lemoine_conjecture For all odd integers $n ≥ 7$ there are prime numbers $p,q$ such that $n = p+2q$ sorry line 39 source
Source documentation
For all odd integers $n ≥ 7$ there are prime numbers $p,q$ such that $n = p+2q$.

depends on sorry what this means
Statement uses: Nat, LT.lt, OfNat.ofNat, Odd, Exists, And, Nat.Prime, Eq, HAdd.hAdd, HMul.hMul
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
14 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem lemoine_conjecture (n : ℕ) (hn : 6 < n) (odd : Odd n) :
    ∃ (p q : ℕ), p.Prime ∧ q.Prime ∧ p + 2 * q = n := by
…
```
theorem Lemoine.lemoine_conjecture_extension For all odd integers $n ≥ 9$ there are odd prime numbers $p,q,r,s$ and natural numbers $a,b$ such that $p+2q = n$, $2+pq = 2^a+r$, $2p+q = 2^b+s$ sorry line 48 source
Source documentation
For all odd integers $n ≥ 9$ there are odd prime numbers $p,q,r,s$ and natural numbers $a,b$ such that $p+2q = n$, $2+pq = 2^a+r$, $2p+q = 2^b+s$

depends on sorry what this means
Statement uses: Nat, LT.lt, OfNat.ofNat, Odd, Exists, And, Lemoine.OddPrime, Eq, HAdd.hAdd, HMul.hMul, HPow.hPow
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
22 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem lemoine_conjecture_extension (n : ℕ) (hn : 8 < n) (odd : Odd n) :
    ∃ (p q r s a b : ℕ), OddPrime p ∧ OddPrime q ∧ OddPrime r ∧ OddPrime s ∧
    p + 2 * q = n ∧ 2 + p * q = 2 ^ a + r ∧ 2 * p + q = 2 ^ b + s := by
…
```
Mentions: Lemoine.OddPrime