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?

Current state (tracked automatically)

All tracked facts and sources

Fact	Value	Source
Formal Conjectures: research statements in the file	2 open, 6 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.

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 2026-11-12}
}

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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_27.equivalent. What is the size of the smallest set $A \subset Z / p Z$ (with at least two elements) for which no element in the sumset $A + A$ has a unique representation?

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.
[CY26] Cao, Jiao-Long, and Ye Yuan. "A Second-Logarithm Lower Bound for Sets with No Unique Sums." arXiv preprint arXiv:2608.06728 (2026).
[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» (2 statements).

theorem green_27.equivalent :
  green_27.equivalent.answer ~[primesAtTop] m

theorem green_27.upper :
    (green_27.upper.answer =o[primesAtTop] upperBest) ∧ (m =O[primesAtTop] green_27.upper.answer)

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 2f91d251c07f. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.