729.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 729

References:

erdosproblems.com/729
[EGRS75] Erdős, P. and Graham, R. L. and Ruzsa, I. Z. and Straus, E. G., On the prime factors of $\binom{2n}{n}$. Math. Comp. (1975), 83-92.
[Er68c] Erdős, P., Aufgabe 557. Elemente Math. (1968), 111-113.

Imports (1)

FormalConjecturesUtil

theorem Erdos729.erdos_729 Let $C>0$ be a constant sorry line 40 source
Source documentation
Let $C>0$ be a constant. Are there infinitely many integers $a,b,n$ with $a+b> n+C\log n$ such that the denominator of $$\frac{n!}{a!b!}$$contains only primes $\ll_C 1$?
Erdős [Er68c] proved that if $a!b!\mid n!$ then $a+b\leq n+O(\log n)$. This has been proved in the affirmative by Barreto and Leeham, using ChatGPT and Aristotle, with a modification of the argument used for [728].

depends on sorry what this means
Statement uses: Iff, True, Real, GT.gt, OfNat.ofNat, Exists, Nat, And, GE.ge, Set.Infinite, Prod, Set.ofPred, Erdos729.erdos_729.match_1, HAdd.hAdd, HMul.hMul, Real.log, Nat.Prime, Eq, padicValNat, Rat.den and 3 more
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
32 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_729 :
    answer(True) ↔ ∀ (C : ℝ) (hC : C > 0),
      ∃ K ≥ 3, Set.Infinite { T : ℕ × ℕ × ℕ |
        let (a, b, n) := T
…
```