369.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 369

References:

erdosproblems.com/369
[ErGr80] Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980).
[EgSe76] Eggleton, R. B. and Selfridge, J. L., Consecutive integers with no large prime factors. J. Austral. Math. Soc. Ser. A (1976), 1--11.
[BFMW20] Bober, J. W. and Fretwell, D. and Martin, G. and Wooley, T. D., Smooth values of polynomials. J. Aust. Math. Soc. (2020), 245--261.
[BaWo98] Balog, Antal and Wooley, Trevor D., On strings of consecutive integers with no large prime factors. J. Austral. Math. Soc. Ser. A (1998), 266-276.

Imports (1)

FormalConjecturesUtil

theorem Erdos369.erdos_369 Let $\epsilon>0$ and $k\geq 2$ sorry line 54 source
Source documentation
Let $\epsilon>0$ and $k\geq 2$. Is it true that, for all sufficiently large $n$, there is a sequence of $k$ consecutive integers in $\{1,\ldots,n\}$ all of which are $n^\epsilon$-smooth?
The problem is trivially true as written (simply taking $\{1,\ldots,k\}$ and $n>k^{1/\epsilon}$). There are (at least) two possible variants which are non-trivial, and it is not clear which Erdős and Graham meant. We formalize the second: each $m\in P$ (where $P$ is the sequence of $k$ consecutive integers sought for) must be in $[n/2,n]$. In this case a positive answer also follows directly from the result of Balog and Wooley [BaWo98] for infinitely many $n$. Proving this is true for all large $n$ does not follow immediately from [BaWo98], but can be deduced using a similar construction, as shown by SkyYang.

depends on sorry what this means
Statement uses: Iff, True, Real, LT.lt, OfNat.ofNat, Nat, LE.le, Filter.Eventually, Exists, And, HDiv.hDiv, HAdd.hAdd, Membership.mem, Finset, Nat.primeFactors, HPow.hPow, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
36 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_369 : answer(True) ↔
    ∀ (ε : ℝ) (hε : 0 < ε) (k : ℕ) (hk : 2 ≤ k),
      ∀ᶠ (n : ℕ) in atTop, ∃ a : ℕ, n / 2 ≤ a + 1 ∧ a + k ≤ n ∧
        ∀ j < k, ∀ p ∈ (a + 1 + j).primeFactors, (p : ℝ) ≤ (n : ℝ) ^ ε := by
…
```