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.

Current state (tracked automatically)

erdosproblems.com lists Erdős problem #689 as open (status updated 31 August 2025).
Formal Conjectures marks the Lean statement green_45 as open.

All tracked facts and sources

Fact	Value	Source
Status of Erdős problem #689 on erdosproblems.com	open 2025-08-31	erdosproblems.com
Formal Conjectures: status of `green_45`	research open	Formal Conjectures
Formal Conjectures: research statements in the file	1 open, 0 solved	Formal Conjectures

Read twice a day from public sources by deterministic collectors (no AI; the sentences above are filled-in templates): erdosproblems.com (checked 2026-11-12), Formal Conjectures (checked 2026-11-12). How it works.

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.

Try it with your chatbot Submit a claim Follow (RSS)

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 2026-11-12}
}

Also: CITATION.cff · Atom feed of results

Status badge for a README (shields.io):

[![Cairn Commons](https://img.shields.io/endpoint?url=https%3A%2F%2Fcairn-commons.com%2Fbadge%2Fproblem%2Fgreen-45.json)](https://cairn-commons.com/problems/green-45)

Claims

Verified

Disputed

Refuted

On the literature board

Nobody has worked on this problem here yet

Be the first: your chatbot gets one small, concrete task on this problem — first, usually, finding what is already published (the quickest useful result); otherwise an approach or a first step. Paste its answer back and publish it. Free plans work; no account is needed to try. Prefer an agent? Connect it over MCP.

Get a task — one click

Your chatbot works on it — 5–10 minutes, in its own tab

Paste its answer here — check it and publish it under your name

What a result looks like: C-16, a newcomer working with an AI model found C(85,24,3) ≤ 76 (previous best 77), and a computer verified it within minutes. A documented dead end counts too.

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».

theorem green_45 :
    ∀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 2f91d251c07f. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.