390.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

3 declarations, 2 contain sorry, 0 declared as axioms

Erdős Problem 390

References:

erdosproblems.com/390
[EGS82] Erdős, P., R. K. Guy, and J. L. Selfridge. "Another Property of 239 and some related questions." Congr. Numer. 34 (1982): 243-257.

Imports (1)

FormalConjecturesUtil

def Erdos390.f Let f n be the smallest integer for which n! can be represented as the product of distinct integers greater than n, the largest of which is f n line 39 source
Source documentation
Let f n be the smallest integer for which n! can be represented as the product of distinct integers greater than n, the largest of which is f n.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: InfSet.sInf, Set.ofPred, Exists, And, StrictMono, LT.lt, OfNat.ofNat, Eq, HSub.hSub, Finset.prod, Finset.Iio, Nat.factorial
10 instances, coercions and auxiliary constants left out.
```
noncomputable def f (n : ℕ) : ℕ := sInf {m : ℕ | ∃ k, ∃ f : ℕ → ℕ, StrictMono f ∧
…
```
theorem Erdos390.erdos_390.variants.theta f n - 2 * n = θ (n / log n) sorry line 44 source
Source documentation
f n - 2 * n = θ (n / log n). This is proved in [EGS82].

depends on sorry what this means
Statement uses: Asymptotics.IsTheta, Nat, Real, Filter.atTop, HSub.hSub, Erdos390.f, HMul.hMul, OfNat.ofNat, HDiv.hDiv, Real.log
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
30 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_390.variants.theta :
    (fun n => f n - 2 * n : ℕ → ℝ) =Θ[atTop] (fun n => n / log (n : ℝ)) := by
…
```
Mentions: Erdos390.f
theorem Erdos390.erdos_390 Does there exists a constant c such that f n - 2 * n ~ c * (n / log n)? sorry line 50 source
Source documentation
Does there exists a constant c such that f n - 2 * n ~ c * (n / log n)?

depends on sorry what this means
Statement uses: Iff, True, Exists, Real, Asymptotics.IsEquivalent, Nat, Filter.atTop, HSub.hSub, Erdos390.f, HMul.hMul, OfNat.ofNat, HDiv.hDiv, Real.log
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
38 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_390 :
    answer(sorry) ↔ ∃ c,
      (fun n => f n - 2 * n : ℕ → ℝ) ~[atTop] (fun n => c * n / log (n : ℝ)) := by
…
```
Mentions: c, Erdos390.f