Ben Green's Open Problem 43

Let N be a large integer. For each prime p with N^0.51 ≤ p < 2N^0.51, pick some residue a(p) ∈ ℤ/pℤ. Is it true that \#n ∈ [N] : n ≡ a(p) pmodp for some p ≫ N^1-o(1)?

Current state (tracked automatically)

Formal Conjectures marks the Lean statement green_43 as open.

All tracked facts and sources

Fact	Value	Source
Formal Conjectures: status of `green_43`	research open	Formal Conjectures
Formal Conjectures: research statements in the file	5 open, 1 solved	Formal Conjectures

Read twice a day from public sources by deterministic collectors (no AI; the sentences above are filled-in templates): 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-43,
  title        = {Ben Green's Open Problem 43},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/green-43}},
  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-43.json)](https://cairn-commons.com/problems/green-43)

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

Let $N$ be a large integer. For each prime $p$ with $N^{0.51} \leq p < 2 N^{0.51}$ , pick some residue $a (p) \in Z / p Z$ . Is it true that $# {n \in [N] : n \equiv a (p) (mod p) for some p} ≫ N^{1 - o (1)} ?$

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.GreensOpenProblems.«43».

theorem green_43 : SievedLowerBoundExp (0.51 : ℝ)

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.

Variants

green_43.variants.alon_peres — [Gr24, Comments to Problem 43; AP92] One of Green's favourite related conjectures: if A ⊆ ℤ/pℤ is a set of size ⌊p/2⌋, does some dilation n A (n an integer)…
green_43.variants.range_N051_N052 — [Gr24, Comments to Problem 43] A minor variant of Problem 43: the same question but with p ranging over N^0.51 ≤ p < N^0.52.
green_43.variants.range_N051_N052_linear — [Gr24, Comments to Problem 43] For the range N^0.51 ≤ p < N^0.52 there may even be a linear lower bound c N.
green_43.variants.general_exponent — [Gr24, Comments to Problem 43] One can replace the exponent 0.51 by any other exponent α: does the lower bound N^1-o(1) hold for sieving primes in [N^α, 2 N^α)…

References

[Gr24] Ben Green's Open Problem 43
[Gr18] Green, Ben. "A note on multiplicative functions on progressions to large moduli." Proc. Roy. Soc. Edinburgh Sect. A 148 (2018), no. 1, 63–77. (Problem 43 is formulated here, [Gr18, Section 4].)
[AP92] Alon, Noga, and Yuval Peres. "Uniform dilations." Geometric and Functional Analysis 2 (1992), no. 1, 1–28. (Source of the favourite variant in [Gr24, Comments to Problem 43].)

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.