473.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

Erdős Problem 473

References:

erdosproblems.com/473
[ErGr80] Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory, Monographies de L'Enseignement Mathématique (1980).

Imports (1)

FormalConjecturesUtil

theorem Erdos473.erdos_473 sorry line 44 source
depends on sorry what this means
Statement uses: Iff, True, Exists, Equiv, Nat, PNat, Nat.Prime, HAdd.hAdd, OfNat.ofNat
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 Erdos473.erdos_473.variants.finite Segal also asked whether for every $n \ge 2$ there is a permutation $a_1, \ldots, a_n$ of $\{1, \ldots, n\}$ such that $a_k + a_{k+1}$ is prime for all $1 \le k < n$ sorry line 54 source
Source documentation
Segal also asked whether for every $n \ge 2$ there is a permutation $a_1, \ldots, a_n$ of $\{1, \ldots, n\}$ such that $a_k + a_{k+1}$ is prime for all $1 \le k < n$. This is conjectured to be true, and has been verified for infinitely many $n$.

depends on sorry what this means
Statement uses: Iff, True, Nat, LE.le, OfNat.ofNat, Exists, Equiv, Fin, LT.lt, HAdd.hAdd, Nat.Prime, Fin.mk
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 erdos_473.variants.finite : answer(sorry) ↔
    ∀ n : ℕ, 2 ≤ n → ∃ a : Fin n ≃ Fin n, ∀ k : Fin n, ∀ h : k.val + 1 < n,
      ((a k).val + 1 + ((a ⟨k.val + 1, h⟩).val + 1)).Prime := by
…
```