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Level A · Machine-checkable Hard Combinatorics P-green-58

Ben Green's Open Problem 58

Suppose A, B ⊆ 1, …, N both have size at least N^0.49. Must the sumset A + B contain a composite number?

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-green-58,
  title        = {Ben Green's Open Problem 58},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/green-58}},
  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

Suppose both have size at least . Must the sumset contain a composite number?

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.GreensOpenProblems.«58». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem green_58 :
    answer(sorry) ↔
      ∀ᶠ (N : ℕ) in Filter.atTop, ∀ᵉ (A ⊆ Finset.Icc 1 N) (B ⊆ Finset.Icc 1 N),
        (N : ℝ) ^ (0.49 : ℝ) ≤ (A.card : ℝ) →
        (N : ℝ) ^ (0.49 : ℝ) ≤ (B.card : ℝ) →
        ∃ m ∈ (A + B), m.Composite

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. Report it as a literature claim.
  • A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

[Ben Green's Open Problem 58](https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf#section.8 Problem 58)

Source and licence

Imported from Formal Conjectures (Ben Green's 100 open problems), commit e6d1743831c2. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.