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Level A · Machine-checkable Hard Number theory P-green-46

Ben Green's Open Problem 46

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

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.

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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 2026-09-28}
}

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Current state

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The problem

The question

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

green_46.improve_upper. We conjecture that the best-known upper bound can be improved.

green_46.improve_upper_conjectured. It seems very likely that we must have [Gr24].

What is the largest for which one may cover the interval by residue classes , one for each prime ?

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.GreensOpenProblems.«46» (3 statements). answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem green_46.improve_lower :
    let ans := (answer(sorry) : ℕ → ℝ)
    (bestLower =o[atTop] ans) ∧ (ans ≪ maxY)
theorem green_46.improve_upper :
    let ans := (answer(sorry) : ℕ → ℝ)
    (ans =o[atTop] bestUpper) ∧ (maxY ≪ ans)
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.

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.