Result 004, Number theory

Hilbert’s tenth problem over ℚ

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.

Disproof or counterexample

The bigger picture

Why it matters

Can a computer always determine whether a polynomial equation has a solution made of fractions? The manuscript claims the answer is no, placing a fundamental limit on automated equation solving over the rational numbers.

What changes?

The unreviewed manuscript reports that no algorithm can take every polynomial with integer coefficients and correctly decide whether it has a rational zero: an assignment of rational numbers to all its variables that makes its value zero. The number of variables is part of the input and is not bounded in advance. This concerns a decision procedure that always terminates, not algorithms for restricted families or particular equations.

What does that help mathematicians do?

If established, the result would rule out any universally terminating test for rational solvability in this setting, regardless of computing power. One can systematically enumerate rational assignments and check them, eventually finding a solution whenever one exists. The claimed impossibility shows that this search cannot be complemented by a procedure guaranteed to recognize every case with no solution. That distinguishes finding witnesses from deciding whether witnesses exist.

Are there practical applications?

Its immediate value is foundational: it would delimit what general-purpose methods for exact rational solutions can promise. It would not make specialized methods useless or show that any particular equation is intractable. For computational number theory, the consequence is a limit on universal guarantees, rather than a runtime estimate or a new solving algorithm.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

2 manuscripts

Hilbert’s tenth problem over the rational numbers

September 24, 2026 62 pages

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.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:Hilberts-tenth-problem-over-the-rational-numbers-September-24-2026,
  author = {{OpenAI}},
  title = {{Hilbert's tenth problem over the rational numbers}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Hilberts-tenth-problem-over-the-rational-numbers-September-24-2026/main.pdf}{OAI:Hilberts-tenth-problem-over-the-rational-numbers-September-24-2026}},
  year = {2026}
}

A pointwise 2-converse for elliptic curves with rational two-torsion

September 24, 2026 91 pages

We prove a pointwise 2-converse for elliptic curves over Q\mathbf Q with nonzero rational two-torsion: if the 2∞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.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:A-pointwise-2-converse-for-elliptic-curves-with-rational-two-torsion-September-24-2026,
  author = {{OpenAI}},
  title = {{A pointwise $2$-converse for elliptic curves with rational two-torsion}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/A-pointwise-2-converse-for-elliptic-curves-with-rational-two-torsion-September-24-2026/paper.pdf}{OAI:A-pointwise-2-converse-for-elliptic-curves-with-rational-two-torsion-September-24-2026}},
  year = {2026}
}

Posts about this result

Aran Nayebi @aran_nayebi, Assistant Professor, Carnegie Mellon

Ok update, yes *this* updates my timelines! github.com/openai/math/blob/main/overview.pdf Rational Hodge over CM abelian varieties is true (032) and BSD for a density-one set (002 & 006). Also Hilbert's tenth problem over Q is false (004), just as everyone expected, but we didn't have a proof for!

Quoting @aran_nayebi: If either Hodge or BSD are proven to be *true* by AI (thereby likely using deep mathematical techniques), then this would update my timelines. This may also mean the Riemann Hypothesis is not far off. But if it's a be...

Oct 6, 2026, 6:40 PM ET

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 9 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.