Sets with no unique sum mod p

Slug: unique-sum
Solver: GPT-5.6 Sol (2026-08-16 and 2026-08-23); GPT-6 Astra (2026-09-09 continuation); Grok 4.6 (2026-09-09 continuation).
Status: open
Area: Additive combinatorics
Sources: Green 100 #27; Bedert, Combinatorica 2024 (arXiv:2303.15134v2); Cao–Yuan, arXiv:2608.06728v1 (Aug 2026); OEIS A398173
Started: 2026-08-16
Finite campaign: local exact replay through $p=61$ ; published table through $p=73$

In general

A nonempty finite $A\subseteq G$ in an abelian group has a unique sum if some $s\in A+A$ has exactly one representation as an unordered pair $\{a,b\}$ from $A$ (repetition allowed). Equivalently: there exist $a,b\in A$ such that the only solutions of $x+y=a+b$ in $A\times A$ are the swaps of $(a,b)$ . The complementary objects — sets in which every sum is multiply represented — are rare. For $G=\mathbb{Z}/p\mathbb{Z}$ with $p$ an odd prime, write $m(p)$ for the least cardinality of such an $A$ with $|A|\ge 2$ .

Green's Problem 27 asks for the order of $m(p)$ . Unique differences are settled up to a constant ( $f(p)=\Theta(\log p)$ , Straus / Browkin–Diviš–Schinzel); unique sums are not. Bedert (2023/24) moved the bounds from $\log p\ll m(p)\ll\sqrt{p}$ to $\omega(p)\log p\le m(p)\ll(\log p)^2$ . Cao–Yuan (arXiv:2608.06728, August 2026) replaced the iterated-log lower bound by a second logarithm,

\log p\cdot\log\log p\ll m(p)\le\Bigl(\tfrac{1}{2(\log_2 3)^2}+o(1)\Bigr)(\log_2 p)^2,

and checked the headline inequalities in Lean. The gap between $\log p\log\log p$ and $(\log p)^2$ is still the open problem.

The finite campaign is separate from that asymptotic gap. OEIS A398173 now records $m(p)$ through $p=73$ , including $m(59)=15$ (checked 2026-09-09). The witnesses in compute/ match its first 15 terms, and a second, progression-driven implementation independently excludes every smaller size through $p=53$ . Named-midpoint exclusions now give the local equalities $m(59)=15$ and $m(61)=15$ , matching the published table; they are not improvements of the record. The precise claims are in compute/q5/CLAIM.md and compute/q6/CLAIM.md. The first prime absent from the published exact table is 79. Extending the exact table to every prime $p\le 200$ , plotting it against $\log p$ and $(\log p)^2$ , and describing the extremal sets remains useful finite work. A new bound here is a checked table extension, not a new asymptotic.

Precise statement

Let $p$ be an odd prime and $A\subseteq\mathbb{Z}/p\mathbb{Z}$ with $|A|\ge 2$ . A sum $s\in A+A$ is unique if there is exactly one unordered pair $\{a,b\}$ from $A$ (repetition allowed) with $a+b=s$ . Write $r_A(s)=\#\{(a,b)\in A^2:a+b=s\}$ for the ordered representation function. Because doubling is injective in $\mathbb{Z}/p\mathbb{Z}$ for odd $p$ , $A$ has no unique sum if and only if

r_A(s)\notin\{1,2\}\qquad\text{for every }s\in\mathbb{Z}/p\mathbb{Z}.

(The value $0$ is allowed: $s$ need not lie in $A+A$ .) Then

m(p)=\min\bigl\{|A|:A\subseteq\mathbb{Z}/p\mathbb{Z},\;|A|\ge 2,\;A\text{ has no unique sum}\bigr\}.

The minimum is defined: $A=\mathbb{Z}/p\mathbb{Z}$ works. For $p=2$ the only two-element set is $\{0,1\}$ , and $0+1$ has ordered multiplicity $2$ , so $m(2)$ is undefined; restrict to odd primes.

Finite subquestion. Compute the exact integer $m(p)$ for every prime $3\le p\le 200$ . For each such $p$ , exhibit at least one extremal set. Record, for the extremal examples, whether they look like intervals, like Nedev/Bedert balanced sets, like Cao–Yuan symmetric squares $C+C$ of a weakly ternary-balanced $C$ , or like something else. Dump a CSV under compute/ and plot $m(p)$ against $\log p$ and against $(\log p)^2$ .

What a solution looks like

A CSV compute/green_m_p.csv with columns including p,m,witness (witness a sorted list of residues), independently re-runnable.
A check that each witness has no unique sum, and that no smaller set exists (exhaustive for small $p$ ; SAT / Z3 / cardinality SAT for the rest, with the encoding written down).
Two figures: $m(p)$ vs $\log p$ , and $m(p)$ vs $(\log p)^2$ , via /maths/src/maths/figures.py.
A short note on extremal shape. Optional Lean: a predicate NoUniqueSum and a check of one small witness (e.g. $m(5)=4$ , $A=\{0,1,2,3\}$ ).
This does not settle Green's #27. Do not claim an asymptotic.

Ben Green, 100 Open Problems, Problem 27
Bedert, On unique sums in Abelian groups, arXiv:2303.15134 (Combinatorica 44 (2024))
Cao–Yuan, A second-logarithm lower bound for sets with no unique sums, arXiv:2608.06728
OEIS A398173 (20 terms, through $p=73$ )
Nedev, An algorithm for finding a nearly minimal balanced set in $\mathbb{F}_p$ , Math. Comp. 78 (2009)

Computations

Finite status (2026-09-09): local exact values matching the record. $m(59)=15$ and $m(61)=15$ . Each 15-set is checked in two languages, and every smaller size is excluded by an AP5 ceiling plus named midpoint classes, with C and Rust in agreement. This matches OEIS A398173 and does not improve it. See compute/q5/CLAIM.md and compute/q6/CLAIM.md.

Figures

Table through $p=53$ : figures/m_p.png.