845.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

1 declarations, 1 contain sorry, 0 declared as axioms

Erdős Problem 845

Reference: erdosproblems.com/845

Imports (1)

FormalConjecturesUtil

theorem Erdos845.erdos_845 Let $C > 0$ sorry line 41 source
Source documentation
Let $C > 0$. Is it true that the set of integers of the form $n = b_1 + \cdots + b_t$, with $b_1 < \cdots < b_t$, where $b_i = 2^{k_i}3^{l_i}$ for $1 \leq i\leq t$ and $b_t \leq Cb_1$ has density $0$?
van Doorn and Everts \cite{vDEv25} have disproved this with $C=6$ - in fact, they prove that all integers can be written as such a sum in which $b_t<6b_1$.
This was formalized in Lean by Alexeev using Aristotle.

depends on sorry what this means
Statement uses: Iff, False, Real, LT.lt, OfNat.ofNat, Prod, Nat, Erdos845.erdos_845.match_1, HMul.hMul, HPow.hPow, Set.HasDensity, Set.ofPred, Exists, Finset, Finset.Nonempty, LE.le, Finset.sup, Finset.inf', Eq, Finset.sum and 1 more
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
44 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_845 :
    answer(False) ↔
      ∀ᵉ (C : ℝ) (hC : 0 < C),
        let f : ℕ × ℕ → ℕ := fun (k, l) ↦ 2 ^ k * 3 ^ l
…
```