234.lean`

Explanation legend Lean4 Domain Explanation Language (L4DEL)

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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

Erdős Problem 234

Reference: erdosproblems.com/234

Imports (1)

FormalConjecturesUtil

theorem Erdos234.erdos_234 Is it true that for all c ≥ 0, the density f c of integers for which (p (n + 1) - p n) / log n < c exists and is a continuous function of c? sorry line 38 source
Source documentation
Is it true that for all c ≥ 0, the density f c of integers for which (p (n + 1) - p n) / log n < c exists and is a continuous function of c?

depends on sorry what this means
Statement uses: Iff, True, Exists, NNReal, Real, And, Continuous, Set.HasDensity, Nat, Set.ofPred, LT.lt, HDiv.hDiv, primeGap, Real.log, Set.univ
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
31 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_234 : answer(sorry) ↔ ∃ f : ℝ≥0 → ℝ, Continuous f ∧
    ∀ c : ℝ≥0, HasDensity {n : ℕ | primeGap n / log n < c} (f c) := by
…
```
Mentions: c, primeGap