1201.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 1201

Reference: erdosproblems.com/1201

Imports (1)

FormalConjecturesUtil

def Erdos1201.Erdos1201Set The set of $n$ for which $P(n(n+1)\cdots(n+k)) > n^{1-\epsilon}$, where $P(m)$ is the greatest prime divisor of $m$ line 36 source
Source documentation
The set of $n$ for which $P(n(n+1)\cdots(n+k)) > n^{1-\epsilon}$, where $P(m)$ is the greatest prime divisor of $m$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real, Nat, Set
Proof uses: Set.ofPred, GT.gt, SupSet.sSup, And, Nat.Prime, Dvd.dvd, Finset.prod, Finset.range, HAdd.hAdd, OfNat.ofNat, HPow.hPow, HSub.hSub
15 instances, coercions and auxiliary constants left out.
```
noncomputable def Erdos1201Set (ε : ℝ) (k : ℕ) : Set ℕ :=
…
```
theorem Erdos1201.erdos_1201 Documented declaration: erdos_1201 sorry line 47 source
Source documentation
Is it true that for every $\epsilon,\eta>0$ there exists a $k$ such that the density of $n$ for which $P(n(n+1)\cdots(n+k))>n^{1-\epsilon}$ is at least $1-\eta$ (where $P(m)$ is the greatest prime divisor of $m$)?

depends on sorry what this means
Statement uses: Iff, True, Real, GT.gt, OfNat.ofNat, EReal, Exists, Nat, GE.ge, Filter.liminf, HDiv.hDiv, Nat.count, Membership.mem, Set, Erdos1201.Erdos1201Set, Classical.propDecidable, Filter.atTop, HSub.hSub
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
53 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_1201 :
    answer(sorry) ↔
      ∀ ε > 0, ∀ η > 0, ∃ k : ℕ,
        atTop.liminf (fun x : ℕ ↦
          (((count (· ∈ Erdos1201Set ε k) x : ℝ) / (x : ℝ)) : EReal)) ≥ (1 - η : EReal) := by
…
```
Mentions: Erdos1201.Erdos1201Set
theorem Erdos1201.erdos_1201.variants.epsilon_half Erdős wrote he could prove this for $\epsilon=1/2$ sorry line 59 source
Source documentation
Erdős wrote he could prove this for $\epsilon=1/2$.

depends on sorry what this means
Statement uses: EReal, GT.gt, OfNat.ofNat, Exists, Nat, GE.ge, Filter.liminf, HDiv.hDiv, Real, Nat.count, Membership.mem, Set, Erdos1201.Erdos1201Set, Classical.propDecidable, Filter.atTop, HSub.hSub
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
60 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_1201.variants.epsilon_half :
    ∀ η > 0, ∃ k : ℕ,
      atTop.liminf (fun x : ℕ ↦
        (((count (· ∈ Erdos1201Set (1 / 2 : ℝ) k) x : ℝ) / (x : ℝ)) : EReal)) ≥ (1 - η : EReal) := by
…
```
Mentions: Erdos1201.Erdos1201Set