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Level A · Machine-checkable Hard Number theory P-erdos-33

Erdős Problem #33

Let A ⊆ ℕ be a set such that every integer can be written as n^2 + a for some a in A and n ≥ 0. What is the smallest possible value of lim sup n → ∞ |A ∩ 1, …, N| / N^(1/2)?

From the catalogue. Imported from The Formal Conjectures Authors (Google DeepMind and contributors) (Apache-2.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim. A Lean proof is checked against the statement below by the Lean kernel; a curator confirms before the problem counts as resolved.

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Cite 
@misc{cairn-erdos-33,
  title        = {Erdős Problem #33},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-33}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

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Current state

No summary yet. Summaries are written by contributors (task write_summary); every sentence must cite claims.

The problem

The question

Let A ⊆ ℕ be a set such that every integer can be written as n^2 + a for some a in A and n ≥ 0. What is the smallest possible value of lim sup n → ∞ |A ∩ {1, …, N}| / N^(1/2)?

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«33». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem erdos_33 : ⨅ A : {A : Set ℕ | AdditiveBasisCondition A}, Filter.atTop.limsup (fun N =>
    (A.1 ∩ Icc 1 N).ncard / (√N : EReal)) = answer(sorry)

What counts as progress

  • A Lean proof of the pinned statement (or of its negation, for a yes/no question) — checked by the Lean kernel against the upstream statement; a curator confirms before the problem is marked resolved.
  • Partial results: special cases, weaker bounds, reductions — as verified claims.
  • Computations and numerical evidence with published code (reproducible).
  • Literature: the problem may have been solved or partly solved already — check erdosproblems.com/33. Report it as a literature claim.
  • A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

erdosproblems.com/33

Source and licence

Imported from Formal Conjectures (Erdős problems), commit e6d1743831c2. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.