∑math20

Number theory · Result 004

Hilbert’s tenth problem over ℚ

2 papers

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.

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▶Introduction: the problem and why it mattersin preparation
▶The presented solutionin preparation

Papers 2

First page of Hilbert’s tenth problem over the rational numbers

Hilbert’s tenth problem over the rational numbers

OpenAI · October 6, 2026

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.

This version updates citations to revised manuscripts in this collection; see the previous version.

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Cite
@misc{OAI:Hilberts-tenth-problem-over-the-rational-numbers-October-6-2026,
  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-October-6-2026/main.pdf}{OAI:Hilberts-tenth-problem-over-the-rational-numbers-October-6-2026}},
  year = {2026}
}
First page of A pointwise 2-converse for elliptic curves with rational two-torsion

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

OpenAI · October 7, 2026

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.

This version updates a citation to a revised manuscript in this collection; see the previous version.

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Cite
@misc{OAI:A-pointwise-2-converse-for-elliptic-curves-with-rational-two-torsion-October-7-2026,
  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-October-7-2026/paper.pdf}{OAI:A-pointwise-2-converse-for-elliptic-curves-with-rational-two-torsion-October-7-2026}},
  year = {2026}
}

Formal verification

No Lean formalization is listed for this result yet. The repository notes that unformalized results could have issues.