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Level A · Machine-checkable Hard Combinatorics P-green-27

Green's Open Problem 27

What is the size of the smallest set A ⊂ ℤ / pℤ (with at least two elements) for which no element in the sumset A + A has a unique representation?

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-27,
  title        = {Green's Open Problem 27},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/green-27}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

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Current state

No summary yet. Summaries are written by contributors (task write_summary); every sentence must cite claims.

The problem

The question

green_27.equivalent. What is the size of the smallest set (with at least two elements) for which no element in the sumset has a unique representation?

green_27.lower. Propose a better lower bound along primes.

green_27.upper. Propose a better upper bound along primes.

References:

  • [Gr24] Green, Ben. "100 open problems." (2024).
  • [Be23] Bedert, Benjamin. "On unique sums in Abelian groups." Combinatorica 44.2 (2024): 269-298.
  • [St76] Straus, E. G. "Differences of residues (mod p)." Journal of Number Theory 8.1 (1976): 40-42.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.GreensOpenProblems.«27» (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_27.equivalent :
  (answer(sorry) : ℕ → ℝ) ~[primesAtTop] m
theorem green_27.lower :
    let ans := (answer(sorry) : ℕ → ℝ)
    (lowerBest =o[primesAtTop] ans) ∧ (ans =O[primesAtTop] m)
theorem green_27.upper :
    let ans := (answer(sorry) : ℕ → ℝ)
    (ans =o[primesAtTop] upperBest) ∧ (m =O[primesAtTop] ans)

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.