23.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

1 declarations, 1 contain sorry, 0 declared as axioms

Green's Open Problem 23

References:

Green, Ben. "100 open problems." (2024).
[FrKlMo25] Frantzikinakis, N., O. Klurman, and J. Moreira. "Partition regularity of Pythagorean pairs." Forum of Mathematics, Pi 13. Cambridge University Press (2025).

Imports (1)

FormalConjecturesUtil

theorem Green23.green_23 Suppose that $\mathbb{N}$ is finitely coloured sorry line 39 source
Source documentation
Suppose that $\mathbb{N}$ is finitely coloured. Are there $x,y$ of the same colour such that $x^2 + y^2$ is a square?
Solved in [FrKlMo25].

depends on sorry what this means
Statement uses: Iff, True, Nat, Fin, Exists, And, LT.lt, OfNat.ofNat, Eq, IsSquare, HAdd.hAdd, HPow.hPow
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
18 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem green_23 :
  answer(True) ↔
    -- For every finite colouring of the natural numbers
    ∀ (k : ℕ) (c : ℕ → Fin k),
    -- there exist two numbers of the same colour whose squares sum to a square
    ∃ x y : ℕ,
      0 < x ∧ 0 < y ∧  -- Exclude trivial x^2 + 0^2 = x^2 solution
      c x = c y ∧      -- Same colour
      IsSquare (x^2 + y^2)
    := by
…
```
Mentions: c