Ben Green's Open Problem 46

We conjecture that the best-known lower bound can be improved.

Current state (tracked automatically)

All tracked facts and sources

Fact	Value	Source
Formal Conjectures: research statements in the file	2 open, 3 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.

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Cite

@misc{cairn-green-46,
  title        = {Ben Green's Open Problem 46},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/green-46}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-12}
}

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Claims

Verified

Disputed

Refuted

On the literature board

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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

green_46.improve_lower. We conjecture that the best-known lower bound can be improved.

green_46.improve_upper_conjectured. It seems very likely that we must have $y ≪ x^{1 + o (1)}$ [Gr24].

What is the largest $y$ for which one may cover the interval $[y]$ by residue classes $a_{p} (mod p)$ , one for each prime $p \leq x$ ?

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.GreensOpenProblems.«46» (2 statements).

theorem green_46.improve_lower :
    (bestLower =o[atTop] green_46.improve_lower.answer) ∧ (green_46.improve_lower.answer ≪ maxY)

theorem green_46.improve_upper_conjectured :
    ∃ o : ℕ → ℝ, (o =o[atTop] fun _ : ℕ ↦ (1 : ℝ)) ∧
      maxY ≪ fun x ↦ (x : ℝ) ^ (1 + o x)

What counts as progress

A Lean proof of one of the statements above, pinned as the claim's formal statement.
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 Open Problem 46
[FGK18] Ford, K., Green, B., Konyagin, S., Maynard, J., & Tao, T. (2018). Long gaps between primes. Journal of the American Mathematical Society, 31(1), 65-105.
[Iw78] Iwaniec, Henryk. "On the problem of Jacobsthal." Demonstratio Mathematica 11.1 (1978): 225-232.
[OAI26a] OpenAI, A quadratic bound for Jacobsthal's function. OpenAI Math Release preprint (2026). https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-quadratic-bound-for-Jacobsthals-function-September-25-2026/paper.pdf
[Ra38] Rankin, R. A., The difference between consecutive prime numbers. J. London Math. Soc. (1938), 242--247.

This is the same problem as Erdős Problem 687.

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.