763.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, 3 contain sorry, 0 declared as axioms

Erdős Problem 763

References:

erdosproblems.com/763
[Er65b] Erdős, Paul, Some recent advances and current problems in number theory. Lectures on Modern Mathematics, Vol. III (1965), 196-244.
[Er70c] Erdős, P., Some problems in additive number theory. Amer. Math. Monthly (1970), 619-621.
[ErFu56] Erdős, P. and Fuchs, W. H. J., On a problem of additive number theory. J. London Math. Soc. (1956), 67-73.
[MoVa90] Montgomery, H. L. and Vaughan, R. C., On the Erdős-Fuchs theorems. (1990), 331-338.

Imports (1)

FormalConjecturesUtil

theorem Erdos763.erdos_763 sorry line 54 source
depends on sorry what this means
Statement uses: Iff, False, Exists, Set, Nat, Real, And, LT.lt, OfNat.ofNat, Asymptotics.IsBigO, Filter.atTop, HSub.hSub, Finset.sum, Finset.range, HAdd.hAdd, AdditiveCombinatorics.sumRep, HMul.hMul
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
34 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Erdos763.erdos_763.variants.erdos_fuchs Erdős and Fuchs [ErFu56] proved that $\sum_{n\leq N} 1_A\ast 1_A(n) = cN+o\left(\frac{N^{1/4}}{(\log N)^{1/2}}\right)$ is impossible sorry line 64 source
Source documentation
Erdős and Fuchs [ErFu56] proved that $\sum_{n\leq N} 1_A\ast 1_A(n) = cN+o\left(\frac{N^{1/4}}{(\log N)^{1/2}}\right)$ is impossible.

depends on sorry what this means
Statement uses: Not, Exists, Set, Nat, Real, And, LT.lt, OfNat.ofNat, Asymptotics.IsLittleO, Filter.atTop, HSub.hSub, Finset.sum, Finset.range, HAdd.hAdd, AdditiveCombinatorics.sumRep, HMul.hMul, HDiv.hDiv, HPow.hPow, Real.sqrt, Real.log
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
50 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_763.variants.erdos_fuchs : ¬ ∃ (A : Set ℕ) (c : ℝ), 0 < c ∧
    (fun N : ℕ ↦ (∑ n ∈ Finset.range (N + 1), sumRep A n : ℝ) - c * N) =o[atTop]
      fun N ↦ (N : ℝ) ^ (1 / 4 : ℝ) / √(Real.log N) := by
…
```
Mentions: c
theorem Erdos763.erdos_763.variants.montgomery_vaughan Jurkat (unpublished) and Montgomery and Vaughan [MoVa90] proved that even $\sum_{n\leq N} 1_A\ast 1_A(n) = cN+o(N^{1/4})$ is impossible sorry line 74 source
Source documentation
Jurkat (unpublished) and Montgomery and Vaughan [MoVa90] proved that even $\sum_{n\leq N} 1_A\ast 1_A(n) = cN+o(N^{1/4})$ is impossible.

depends on sorry what this means
Statement uses: Not, Exists, Set, Nat, Real, And, LT.lt, OfNat.ofNat, Asymptotics.IsLittleO, Filter.atTop, HSub.hSub, Finset.sum, Finset.range, HAdd.hAdd, AdditiveCombinatorics.sumRep, HMul.hMul, HPow.hPow, HDiv.hDiv
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
50 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_763.variants.montgomery_vaughan : ¬ ∃ (A : Set ℕ) (c : ℝ), 0 < c ∧
    (fun N : ℕ ↦ (∑ n ∈ Finset.range (N + 1), sumRep A n : ℝ) - c * N) =o[atTop]
      fun N ↦ (N : ℝ) ^ (1 / 4 : ℝ) := by
…
```
Mentions: c