---
title: "Hilbert’s tenth problem over ℚ"
authors:
  - "OpenAI (internal model)"
year: 2026
venue: "openai/math manuscript collection (GitHub), result family 004; unreviewed"
arxiv: null
doi: null
source: "https://github.com/openai/math/blob/main/preprints/Hilberts-tenth-problem-over-the-rational-numbers-September-24-2026/main.pdf"
topics:
  - "openai-math"
  - "ai-generated-math"
  - "number-theory"
  - "unformalized"
seed_rank: 1864
seed_batch: "openai-math-2026-10-07"
reviewed: "2026-10-07"
pool: "maths-foundations"
relevance_score: 7
lineage: ai-mathematical-reasoning
cites:
  - title: "Hilbert’s tenth problem over the rational numbers"
    url: "https://github.com/openai/math/blob/main/preprints/Hilberts-tenth-problem-over-the-rational-numbers-September-24-2026/main.pdf"
    year: 2026
    arxiv: null
    doi: null
  - title: "A pointwise 2-converse for elliptic curves with rational two-torsion"
    url: "https://github.com/openai/math/blob/main/preprints/A-pointwise-2-converse-for-elliptic-curves-with-rational-two-torsion-September-24-2026/paper.pdf"
    year: 2026
    arxiv: null
    doi: null
---

# Hilbert’s tenth problem over ℚ

## One-sentence takeaway

OpenAI's result family 004 (Number theory) claims: Proves that no algorithm decides whether an integer-coefficient polynomial in an arbitrary number of variables has a rational zero, resolving Hilbert's tenth problem over ℚ negatively.

## Why it matters here

Pure mathematics with no direct engine use; it is in the library as part of the openai/math corpus, which records what an AI research model claims to have proved. More broadly, every card in this batch is evidence for the `ai-mathematical-reasoning` thread: per the repo README, an unreleased internal OpenAI model produced it, using about three hours of ChatGPT Pro thinking compute per result on average, across roughly 4,000 posed problems.

## Key ideas

- *Hilbert’s tenth problem over the rational numbers*: We give a negative answer to Hilbert's tenth problem over the rational numbers: no algorithm decides whether a polynomial with integer coefficients has a rational zero. The number of variables is part of the input.
- *A pointwise 2-converse for elliptic curves with rational two-torsion*: We prove a pointwise 2-converse for elliptic curves over $\mathbf Q$ with nonzero rational two-torsion: if the $2^\infty$-Selmer corank is zero or one, then the analytic rank and Mordell–Weil rank equal that corank, and the Shafarevich–Tate group is finite. The result allows arbitrary reduction at 2.

## Caveats

- Unreviewed: these are AI-generated manuscripts posted to GitHub with no peer review or arXiv deposit. The repo README says results are "at different stages of verification" and that "some of the unformalized results could have issues."
- This card restates the repo's family summary, manuscript abstracts and Lean scope notes. It is not an independent check of the proofs, and the scope of each theorem is exactly as stated there.
- Lean: not formalized. The repo has no Lean development for this family, so the claim rests on the prose manuscript(s) alone.

## Links

- Manuscript: [Hilbert’s tenth problem over the rational numbers](https://github.com/openai/math/blob/main/preprints/Hilberts-tenth-problem-over-the-rational-numbers-September-24-2026/main.pdf)
- Manuscript: [A pointwise 2-converse for elliptic curves with rational two-torsion](https://github.com/openai/math/blob/main/preprints/A-pointwise-2-converse-for-elliptic-curves-with-rational-two-torsion-September-24-2026/paper.pdf)
- Family entry: https://github.com/openai/math/blob/main/CONTENTS.md
- Overview PDF: https://github.com/openai/math/blob/main/overview.pdf
- Repo: https://github.com/openai/math