Ben Green's Open Problem 45

Can we pick residue classes a_p pmodp, one for each prime p ≤ N, such that every integer ≤ N lies in at least 10 of them? Erdős remarks that he does not know how to answer it with 10 replaced by 2; this is Erdos689.erdos_689.

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-45,
  title        = {Ben Green's Open Problem 45},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/green-45}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

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Disputed

Refuted

On the literature board

The problem

The question

Can we pick residue classes $a_{p} (mod p)$ , one for each prime $p \leq N$ , such that every integer $\leq N$ lies in at least $10$ of them?

Erdős remarks that he does not know how to answer it with $10$ replaced by $2$ ; this is Erdos689.erdos_689.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.GreensOpenProblems.«45». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem green_45 :
    answer(sorry) ↔ ∀ᶠ N in .atTop, ∃ a : ℕ → ℕ, ∀ m ∈ Finset.Icc 1 N,
      10 ≤ (Finset.Icc 1 N |>.filter fun p => p.Prime ∧ a p ≡ m [MOD p]).card

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

The version of this problem with $10$ replaced by $2$ is Erdos689.erdos_689 in FormalConjectures.ErdosProblems.«689».

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.