Green's Open Problem 9

Problem 9 (ii): is r_5(N) ≪ N(log N)^-c?

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-9,
  title        = {Green's Open Problem 9},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/green-9}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

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Claims

Verified

Disputed

Refuted

On the literature board

The problem

The question

green_9_ii. Problem 9 (ii): is $r_{5} (N) ≪ N (lo g N)^{- c}$ ?

green_9_iii. Problem 9 (iii): is $r_{4} (F_{5}^{n}) ≪ N^{1 - c}$ , where $N = 5^{n}$ ?

References:

[Gr24] Green, Ben. "100 open problems." (2024).
[BlSi20] Bloom, Thomas F., and Olof Sisask. "Breaking the logarithmic barrier in Roth's theorem on arithmetic progressions." arXiv preprint arXiv:2007.03528 (2020).

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.GreensOpenProblems.«9» (2 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_9_ii : answer(sorry) ↔
    ∃ c > (0 : ℝ), (fun (N : ℕ) ↦ (r 5 N : ℝ))
      ≪ fun (N : ℕ) ↦ (N : ℝ) * (Real.log N) ^ (-c)

theorem green_9_iii : answer(sorry) ↔
    ∃ c > (0 : ℝ), (fun (n : ℕ) ↦ ((Finset.univ : Finset (𝔽₅ n)).maxAPFreeCard 4 : ℝ))
      ≪ fun (n : ℕ) ↦ ((5 : ℝ) ^ n) ^ (1 - c)

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.

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.