728.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

Erdős Problem 728

Reference: erdosproblems.com/728

Imports (1)

FormalConjecturesUtil

theorem Erdos728.erdos_728 Let $\varepsilon$ be sufficiently small and $C, C' > 0$ sorry line 51 source
Source documentation
Let $\varepsilon$ be sufficiently small and $C, C' > 0$. Are there integers $a, b, n$ such that $$a, b > \varepsilon n\quad a!\, b! \mid n!\, (a + b - n)!, $$ and $$C \log n < a + b - n < C' \log n ?$$
Note that the website currently displays a simpler (trivial) version of this problem because $a + b$ isn't assumed to be in the $n + O(\log n)$ regime.
Barreto and ChatGPT-5.2 have proved that, for any $0 < C_1 < C_2$, there are infinitely many $a, b, n$ with $b = n/2$, $a = n/2 + O(\log n)$, and $C_1 \log n < a + b - n < C_2 \log n$ such that $a! b! \mid n! (a + b - n)!$
This appears to answer the question in the spirit it was intended.
This was formalized in Lean by Alexeev using Aristotle.

depends on sorry what this means
Statement uses: Iff, True, Filter.Eventually, Real, GT.gt, OfNat.ofNat, Exists, Nat, And, LT.lt, HMul.hMul, Dvd.dvd, Nat.factorial, HSub.hSub, HAdd.hAdd, Real.log, nhdsWithin, Set.Ioi
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
40 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_728 :
    answer(True) ↔
      ∀ᶠ ε : ℝ in 𝓝[>] 0, ∀ C > (0 : ℝ), ∀ C' > C,
        ∃ a b n : ℕ,
          0 < n ∧
          ε * n < a ∧
          ε * n < b ∧
          a ! * b ! ∣ n ! * (a + b - n)! ∧
          a + b > n + C * log n ∧
          a + b < n + C' * log n := by
…
```