Green's Open Problem 44

Sieve [N] by removing half the residue classes mod p_i, for primes 2 leqslant p_1 < p_2 < … < p_1000 < N^9/10. Does the remaining set have size at most 1/10 N? We interpret "half the residue classes" as ⌊ p_i / 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-green-44,
  title        = {Green's Open Problem 44},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/green-44}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

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Claims

Verified

Disputed

Refuted

On the literature board

The problem

The question

Sieve $[N]$ by removing half the residue classes mod $p_{i}$ , for primes $2 ⩽ p_{1} < p_{2} < \dots < p_{1000} < N^{9/10}$ . Does the remaining set have size at most $\frac{1}{10} N$ ?

We interpret "half the residue classes" as $⌊ p_{i} /2 ⌋$ .

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.GreensOpenProblems.«44». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem green_44 :
    answer(sorry) ↔ ∀ (N : ℕ) (p : Fin 1000 → ℕ) (A : (i : Fin 1000) → Finset (ZMod (p i))),
      let remaining := (Finset.Icc 1 N).filter (fun x => ∀ i, (x : ZMod (p i)) ∉ A i)
      (∀ i, (p i).Prime) →
      StrictMono p →
      (p 999) ^ 10 < N ^ 9 →
      (∀ i, (A i).card = (p i) / 2) →
      10 * remaining.card ≤ N

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

[Gr24] Ben Green's 100 Open Problems
[Er80] Erdős, Paul. "A survey of problems in combinatorial number theory." Annals of Discrete Mathematics 6 (1980): 89-115.

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.