GoldbachConjecture.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

2 declarations, 2 contain sorry, 0 declared as axioms

Goldbach's conjecture

References:

Imports (1)

FormalConjecturesUtil

theorem GoldbachConjecture.goldbach Goldbach's conjecture sorry line 36 source
Source documentation
Can every even integer greater than 2 be written as the sum of two primes?
L4DEL explanation
Can every even number greater than 2 be written as a sum of two primes? The file marks this as open.
1. The statement is "answer(sorry) iff for every even n greater than 2 there are primes p and q with n = p + q".
2. The attribute reads category research open and the proof body is sorry.
See also: TernaryGoldbachConjecture.ternaryGoldbach
Sources and attribution: Pinned Lean source (1646ca16afd6)
depends on sorry what this means
Statement uses: Iff, True, Nat, LT.lt, OfNat.ofNat, Even, Exists, And, Prime, Eq, HAdd.hAdd
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
10 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem goldbach :
    answer(sorry) ↔ ∀ n : ℕ, 2 < n → Even n → ∃ p q, Prime p ∧ Prime q ∧ n = p + q := by
…
```
theorem TernaryGoldbachConjecture.ternaryGoldbach Ternary Goldbach sorry line 52 source
Source documentation
Can every odd integer greater than 5 be written as the sum of three primes? (A prime may be used more than once.)
NB. While Harald Helfgott's solution is not published in a peer-reviewed journal yet, his results seem generally accepted.
L4DEL explanation
Every odd number greater than 5 is a sum of three primes (primes may repeat); the file marks this as solved.
1. The attribute reads category research solved, and the docstring credits Harald Helfgott while noting the solution is not yet peer-reviewed in a journal.
2. The statement is a plain theorem with no answer slot; the proof body here is sorry.
See also: GoldbachConjecture.goldbach
Sources and attribution: Pinned Lean source (1646ca16afd6)
depends on sorry what this means
Statement uses: Nat, LT.lt, OfNat.ofNat, Odd, Exists, And, Nat.Prime, Eq, HAdd.hAdd
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
10 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem ternaryGoldbach (n : ℕ) (hn : 5 < n) (hn_odd : Odd n) :
    ∃ p q r, Nat.Prime p ∧ Nat.Prime q ∧ Nat.Prime r ∧ n = p + q + r := by
…
```