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Math-2331 · mathematics · Number theory

Single-Solution Bounds for Diophantine Equations

Status Open · unsolvedmath.comLLM status Open unverified

Statement

Definitions and mathematical statement

Shared notation.
$\mathbb N$, $\mathbb Z$, $\mathbb Q$, $\mathbb R$, and $\mathbb C$ denote the natural numbers, integers, rationals, reals, and complex numbers, respectively. All additional parameters and hypotheses are specified in the question.
Statement recorded in the supplied source.
Can Matiyasevich’s theorem for exponential Diophantine equations—an equation with at most one solution in the variables for each parameter, while some parameter has a solution exceeding every prescribed total effectively computable bound—be improved to genuine Diophantine equations without exponentiation, possibly allowing finitely many rather than at most one solution?

Formulation references: [S1], [S2].

Short English statement

Can Matiyasevich’s theorem for exponential Diophantine equations—an equation with at most one solution in the variables for each parameter, while some parameter has a solution exceeding every prescribed total effectively computable bound—be improved to genuine Diophantine equations without exponentiation, possibly allowing finitely many rather than at most one solution?

Sources

Catalogue entry

Statement licence: CC-BY-SA-4.0 · Source file

Contributions 1

Math-2331-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2331

The theorem has unique natural-number witnesses, so undecidable recursively enumerable sets preclude a computable bound for all such witnesses. The problem's witness domain is not specified precisely enough to identify every requested single-solution clause with that theorem, and complete coverage is not established here.

Single-Solution Bounds for Diophantine Equations

OpenAI's claim

OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 242; 2026-09-24), the asserted result is that every recursively enumerable relation on natural-number tuples has an integer-polynomial representation with exactly one complete auxiliary witness for each member.

Mathematical statement of the claimed result

Manuscript 1.
Use $\mathbb N$ including zero. For each $n\ge1$ and recursively enumerable $S\subseteq\mathbb N^n$, the claim gives an integer $m\ge1$ and an integral polynomial $P(a,w)$ with \[ \forall a\in\mathbb N^n:\quad |\{w\in\mathbb N^m:P(a,w)=0\}|= \begin{cases}1,&a\in S,\\0,&a\notin S.\end{cases} \] The number of witness variables is fixed for the set $S$. This asserts a unique Diophantine witness for every member, rather than merely a finite number of witnesses.

Outline of the argument

For manuscript 1 (family 242; 2026-09-24), pell equations recover exponentiation, but their indices are selected using a uniquely coded rank-one elliptic curve. Bounded height error gives an exact growth cutoff, and canonical arithmetic codes then encode each halting computation with a unique full tuple.

Formal verification

For manuscript 1 (family 242; 2026-09-24), OpenAI lists Lean statements for this assertion: every recursively enumerable relation on natural-number tuples has an integer-polynomial representation with exactly one complete auxiliary witness for each member.

The selected statement includes the clause used for this problem.

The linked family Comparator checks are SingleFold.lean.

Their named declarations include OAI.SingleFold.main.

This site has not executed the Lean code. Described statement coverage is not an independent proof check.

Status caveat

The manuscripts were produced by an internal OpenAI model. OpenAI cautions that results without formalization can contain mistakes. This site has not independently checked the proofs; the label records the scope of OpenAI's claim.

Sources

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