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Hilbert's tenth problem over the rational numbers

Hilbert's tenth problem (1900) asked for a procedure deciding whether a polynomial equation with integer coefficients has an integer solution; Davis, Putnam, Robinson and Matiyasevich showed by 1970 that none exists. Over Q\mathbb Q the question remained open: enumeration finds any rational zero, but no method certifies that none exists, and the natural transfer route, an existential definition of Z\mathbb Z in Q\mathbb Q, would contradict Mazur's conjecture on the topology of rational points. Is there an algorithm which, given f∈Z[X1,…,Xn]f\in\mathbb Z[X_1,\ldots,X_n] with nn part of the input, decides whether ff has a zero in Qn\mathbb Q^n?

Result
Disproved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Argument
Field
Diophantine decidability and definability, arithmetic of elliptic curves
Posed by
David Hilbert (1900) for the integer problem; the rational version has been the central open case since the Davis-Putnam-Robinson-Matiyasevich theorem (1970)
Year posed
—
Years open
—
Solved
2026-09-24
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Unreviewed
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
70 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Claims a negative answer: rational solvability of integer polynomial equations (variables unbounded) is undecidable, of Turing degree 0′0', and stays so for total degree at most four. The method reduces integer solvability to finitely many rational-solvability tests per input, using rank-one elliptic curves, a five-point height estimate and a parity condition from 2-descent. It does not define Z\mathbb Z existentially in Q\mathbb Q (consistent with Mazur's conjecture), does not bound the number of variables, and does not address other number fields beyond what is known.

What the AI did

The OpenAI math release (github.com/openai/math) states that its results were produced by an unreleased internal OpenAI model, with on average about three hours of ChatGPT Pro thinking compute per result, under one fixed procedure applied to roughly 4,000 posed problems; outputs were then grouped into families and filtered for significance. The manuscript is credited to OpenAI alone and names no human author. The proof relies on two other manuscripts from the same release as theorem inputs: a pointwise 2-converse (the family companion) and Fontaine-Mazur modularity at the prime 2 (a separate family).

Verification

No independent mathematician has checked this yet. Checked here: Theorem 1.1 of the TeX source read against the posed problem; it states there is no algorithm deciding rational solvability of integer polynomials with the number of variables part of the input, which is the problem as posed. A corollary gives Turing degree 0′0', also for quartics and sums of squares of quadrics. The paper does not give an existential definition of Z\mathbb Z in Q\mathbb Q; it uses a finite-test reduction with compactness. Its inputs include two unrefereed manuscripts of the same release (pointwise 2-converse; Fontaine-Mazur modularity at 2), so the result is only as solid as they are. No Lean formalization is listed.

Sources

Changelog1 change

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