Ben Green's Open Problem 62

Let p be a large prime, and let A be the set of all primes less than p. Is every x ∈ 1, …, p-1 congruent to some product a_1 a_2 where a_1, a_2 ∈ A?

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.

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Cite

@misc{cairn-green-62,
  title        = {Ben Green's Open Problem 62},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/green-62}},
  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

Let $p$ be a large prime, and let $A$ be the set of all primes less than $p$ . Is every $x \in {1, \dots, p - 1}$ congruent to some product $a_{1} a_{2}$ where $a_{1}, a_{2} \in A$ ?

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.GreensOpenProblems.«62». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem green_62 :
    answer(sorry) ↔
      ∀ᶠ p in atTop, p.Prime →
        let A := (Finset.range p).filter Nat.Prime
        ∀ x : ℕ, 1 ≤ x ∧ x < p →
          ∃ a₁ ∈ A, ∃ a₂ ∈ A, (x : ZMod p) = (a₁ * a₂ : ZMod p)

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

Ben Green's Open Problem 62

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.