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Hilbert’s tenth problem over ℚ

OpenAI groups related papers into a “family”. OpenAI’s catalogue describes this one:

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.

The release has no Lean statement for this family, so there is no formal statement to compare with the result described above.

Two current papers; two replaced by later editions.

What the papers state

Theorem 1.1

In Hilbert’s tenth problem over the rational numbers, 6 October 2026

Possible match to OpenAI’s description, selected from the papers’ main statements because it shares the most wording. No review confirming the selection is recorded.

There is no algorithm which, given a polynomial f\in\Z[X_1,\ldots,X_n], with n part of the input, decides whether f has a zero in \Q^n.

Theorem 1.1 (The rational-two-torsion 2-converse)

In A pointwise 2-converse for elliptic curves with rational two-torsion, 7 October 2026

The paper’s main theorem is shown as a possible match to OpenAI’s description. No review confirming the selection is recorded.

Let E/\Q be an elliptic curve with E(\Q)[2]\ne0 and s_2(E)\in\{0,1\}. Then

\ord_{s=1}L(E,s)=\rank_{\Z}E(\Q)=s_2(E), \qquad \Sha(E/\Q)\text{ is finite}.

The release does not say which paper theorem its catalogue describes. Both statements above are shown whole, the closer match to the description’s wording first. The atlas has not reviewed either choice.

Reported proof verification
The release has no Lean statement for this family. The papers are available to read and assess.
Papers and explanations
Two current papers, each readable here as a page beside its PDF.

Papers

Each paper opens as a reading page in the browser, with the PDF beside it. The statements each paper opens with are on its reading page.

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

7 October 2026, current edition.

Read it herePDF

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

Earlier editions: 24 September 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.

Previous editions

Lean statements

The release has no Lean statement for this family. Its papers remain available to read, and contributors can propose a formal statement of a result. Suggest a change

Independent comparisons and proof-verification reports

Does the Lean express the advertised result?

The release contains no Lean statement for this family, so there is no formal statement to compare with OpenAI’s description.

What these papers cite in the release, and what cites them

These links come from the papers’ bibliographies. A citation records that one paper cites another; how each cited result is used has not been examined here.

Within this family

Papers elsewhere in the release that cite this family’s papers

References outside the release

Each paper’s outside references are listed with it on its reading page, where each citation is placed. Every outside work cited in the release: References outside the release

Explain a result, compare statements or propose a correction

Atlas has no contributed explanation for this family yet. You can offer one with a GitHub account. You receive named credit and disclose any AI assistance.

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What people have said

Commentary on this family by others, dated and credited. It is commentary, not a finding.

  • An unsigned public tier list, the OpenAI Math Tier List, classes it as a proof and rates its significance, if it holds up, as “Would win a Fields Medal”. The page says that nothing on it is verified.

    Author not stated. Source

  • Scientific American (byline not recorded), The most exciting claims from OpenAI’s heap of new proofs, 8 October 2026. Source

    Pasten says the Hilbert’s tenth problem over the rationals proof combines two of their own ideas from different areas in a way they had not foreseen and calls the overall strategy original.

  • tparker with nine answerers, How many of the results released by OpenAI went against the conventional mathematical wisdom?, 8 October 2026. Source

    Sawin thinks most experts expected a negative answer for Hilbert’s tenth problem over Q, though no conjecture implied it.

  • Kevin Buzzard, Algebraic number theory results without Lean, 7 October 2026. Source

    An expert flags that the release gives no Lean for Hilbert’s tenth problem over the rationals and suggests it may be the last mile of a programme already close to done rather than a new idea.

Questions awaiting review

Each item identifies a statement, definition or report that needs further review. Its link opens a GitHub form with the source version already recorded.

Identify the theorem the catalogue describes

  • Family 004 The theorem shown was selected from shared wording with OpenAI’s description. Atlas records no review confirming that it is the theorem the description means. Open the form

State the result in Lean

  • Family 004 The release has no Lean statement for this family. Open the form