391.lean`

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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

3 declarations, 2 contain sorry, 0 declared as axioms

Erdős Problem 391

References:

erdosproblems.com/391
[ErGr80] Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980).
[Er96b] Erdős, Paul, Some problems I presented or planned to present in my short talk. Analytic number theory, Vol. 1 (Allerton Park, IL, 1995) (1996), 333-335.
[AlGr77] Alladi, Krishnaswami and Grinstead, Charles, On the decomposition of $n!$ into prime powers. J. Number Theory (1977), 452-458.
[ACRSTUV25] B. Alexeev, E. Conway, M. Rosenfeld, A. Sutherland, T. Tao, M. Uhr, and K. Ventullo, Decomposing a factorial into large factors. arXiv:2503.20170 (2025).

Imports (1)

FormalConjecturesUtil

def Erdos391.t t n is the largest t such that n! is a product of n factors all at least t, i.e. the maximal line 43 source
Source documentation
t n is the largest t such that n! is a product of n factors all at least t, i.e. the maximal $a_1$ in a representation $n! = a_1 \cdots a_n$ with $a_1 \leq \cdots \leq a_n$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: SupSet.sSup, Set.ofPred, Exists, Fin, And, Eq, Finset.prod, Finset.univ, Nat.factorial, LE.le
4 instances, coercions and auxiliary constants left out.
```
noncomputable def t (n : ℕ) : ℕ :=
…
```
theorem Erdos391.erdos_391 sorry line 69 source
depends on sorry what this means
Statement uses: Iff, True, Filter.Tendsto, Nat, Real, HDiv.hDiv, Erdos391.t, Filter.atTop, nhds, OfNat.ofNat, Real.exp
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
23 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Erdos391.erdos_391.variants.deficit sorry line 83 source
depends on sorry what this means
Statement uses: Iff, True, Exists, Real, And, LT.lt, OfNat.ofNat, Set.Infinite, Nat, Set.ofPred, LE.le, HDiv.hDiv, Erdos391.t, HSub.hSub, Real.exp, Real.log
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
27 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.