438.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, 1 contain sorry, 0 declared as axioms

Erdős Problem 438

References:

erdosproblems.com/438
[Er80] Erdős, Paul, A survey of problems in combinatorial number theory. Ann. Discrete Math. (1980), 89-115.
[Er80c] Erdős, Paul, Nine little known problems in combinatorial number theory. Normat (1980), 155-164, 180.
[ErGr80] Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980).
[LOS83] Lagarias, J. C. and Odlyzko, A. M. and Shearer, J. B., On the density of sequences of integers the sum of no two of which is a square. II. General sequences. J. Combin. Theory Ser. A (1983), 123-139.
[KLS02] Khalfalah, A. and Lodha, S. and Szemerédi, E., Tight bound for the density of sequence of integers the sum of no two of which is a perfect square. Discrete Math. (2002), 243-255.

Imports (1)

FormalConjecturesUtil

def Erdos438.SquareSumFree A finite set of natural numbers is square-sum-free if the sum of any two of its elements (possibly equal) is not a square line 48 source
Source documentation
A finite set of natural numbers is square-sum-free if the sum of any two of its elements (possibly equal) is not a square.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Finset, Nat
Proof uses: Membership.mem, Not, IsSquare, HAdd.hAdd
5 instances, coercions and auxiliary constants left out.
```
def SquareSumFree (A : Finset ℕ) : Prop :=
…
```
def Erdos438.extremalSize The largest size of a square-sum-free subset of {1, …, N} line 52 source
Source documentation
The largest size of a square-sum-free subset of {1, …, N}.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Finset.sup, Finset, Finset.filter, Erdos438.SquareSumFree, Classical.propDecidable, Finset.powerset, Finset.Icc, OfNat.ofNat, Finset.card
6 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
```
noncomputable def extremalSize (N : ℕ) : ℕ := by
…
```
theorem Erdos438.erdos_438 sorry line 68 source
depends on sorry what this means
Statement uses: Filter.Tendsto, Nat, Real, HDiv.hDiv, Erdos438.extremalSize, Filter.atTop, nhds, OfNat.ofNat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
26 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.