OpenAI claims Hilbert’s tenth problem over ℚ is undecidable
Awaiting review
Importance: historic (5 of 5)The takeaway
Family 004 of OpenAI’s 6 Oct 2026 release claims that no algorithm decides whether an integer polynomial has a rational zero, the rational-number version of Hilbert’s tenth problem, open since Matiyasevich settled the integer case in 1970. The 62-page proof avoids defining the integers inside the rationals; it reduces integer solvability to infinitely many finite rational tests. It is not formalized and leans on three companion manuscripts. Frank Calegari said it ‘threw me for a loop’, and Hector Pasten called the strategy ‘original’.
Status
- Claim
Awaiting review
- Our reporting
- Medium confidence
- Verification
- Preprint only
- Importance
- Historic (5 of 5)
- Last verified
- 29 October 2026
Your AI and this story
- GPT-6 Astra159 days after its cutoff
- Claude Opus 5.598 days after its cutoff
- Gemini 4 Argon189 days after its cutoff
- Grok 4.7128 days after its cutoff
None of these four assistants can know about it. The closest, Claude Opus 5.5, stops 98 days before it.
Key facts
- Theorem 1.1 (verbatim): ‘There is no algorithm which, given a polynomial f ∈ Z[X1, . . . , Xn], with n part of the input, decides whether f has a zero in Qn.’ Paper: 62 pages, version of 6 Oct (citation update of the 24 Sep original); companion ‘A pointwise 2-converse for elliptic curves with rational two-torsion’ (91 pages, 7 Oct version)
- Strategy (paper, §1.2): for each nonconstant f, ‘an effectively generated sequence of finite tests’ answerable by rational-solvability queries, with ‘f has an integer zero ⇐⇒ every test succeeds’. Ingredients: compactness, a rank-one elliptic group whose point indices encode integers, local comparisons of elliptic logarithms, and ‘a height bound from odd contact with five points’ proved through a family of abelian surfaces and modularity
- Why the obvious route was closed (paper, §1.1): Mazur conjectured that the real closure of the rational points of a variety has finitely many components, which ‘would rule out a Diophantine definition of Z in Q’; Koenigsmann’s universal definition of ℤ and Poonen’s results on subrings ℤ[S⁻¹] did not give the existential definition. The number-field case was finished by Koymans–Pagano and by Alpöge–Bhargava–Ho–Shnidman, but ℚ is not a finitely generated ℤ-algebra
- Dependencies inside the release: the paper cites the pointwise 2-converse, ‘Fontaine–Mazur modularity at the prime 2’ (family 010) and the Goldfeld/2-converse manuscript (family 006); ‘Its modularity step uses the odd regular Fontaine–Mazur theorem at 2’. None of these is formalized
- Frank Calegari (Persiflage, 8 Oct, verbatim): ‘This one threw me for a loop.’ ‘As for Hilbert’s tenth problem itself, it seemed plausible to me that it was actually solvable over the rational numbers. I don’t think I would have put money on that, however.’ He still hopes the problem is decidable for curves, which ‘should be a consequence of conjectures that we believe’
- Hector Pasten (Scientific American, 8 Oct): the proof uses two of his ideas from different areas, and ‘even he hadn’t realized they could be put together in this way’. Verbatim: ‘It’s extremely surprising to me that they connect … Their general strategy is original.’
- Bill Gasarch (Computational Complexity blog, 11 Oct): ‘OpenAI showed that this problem is also undecidable. It has not been Lean-formalized. Judging from the table of contents, the proof looks like it uses deep math.’
- Follow-up: Zhi-Wei Sun, ‘86 unknowns for Hilbert’s tenth problem over the rationals’ (arXiv 2610.20691, 14 Oct): ‘Recently, Hilbert’s Tenth Problem over the field Q of rational numbers was solved negatively by OpenAI based on a five-contact height estimate and a pole-parity construction’; he proves undecidability already for 86 rational unknowns
Show 2 more
- Questions raised: on Calegari’s blog Maarten Derickx (10 Oct) pointed to a Facebook post by Jennifer Park saying the paper cites her work ‘in a completely wrong way’; an anonymous reply said the point ‘doesn’t seem at all a cogent point’ and that the ideas need study. We did not see Park’s post. The paper cites Eisenträger–Miller–Park–Shlapentokh on subrings of ℚ
- Felipe Voloch (comment, 11 Oct) listed conjectural routes to decidability for curves of genus > 1 (ABC/Vojta with explicit constants, the section conjecture, Tate + Hodge + Fontaine–Mazur) and finiteness of Sha for genus 1
What happened
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 in 1970 that none exists. The same question with rational solutions has been open ever since. The natural route, defining the integers inside the rationals by a polynomial condition, is expected to fail: Mazur’s conjecture on the topology of rational points would rule it out.
OpenAI’s manuscript takes a different route. For each polynomial it generates an infinite list of finite tests, each answerable with finitely many questions about rational solvability, such that the polynomial has an integer zero exactly when every test passes. A rational-solvability algorithm would then decide integer solvability, which is impossible. The arithmetic work is to show that some test fails whenever there is no integer zero. For this the paper encodes integers as multiples of a point on a rank-one elliptic curve and controls heights through a five-point covering construction, abelian surfaces and modularity. That modularity step uses the release’s own Fontaine–Mazur theorem at 2, and the argument also cites the release’s 2-converse papers.
Why it matters
A negative answer over ℚ would close one of the best-known questions left over from Hilbert’s list, and it would do so without the existential definition of ℤ that most experts thought unavailable. Because the proof is long, unformalized and depends on other unrefereed manuscripts in the same release, it should be treated as a serious claim awaiting expert verification.
What is disputed or not yet verified
| Verification | Unrefereed manuscript; no Lean formalization; depends on unverified companion manuscripts (families 006, 010) |
|---|
Reactions
Specialists were surprised. Frank Calegari wrote that the result “threw me for a loop” and that he had thought the problem might be decidable over ℚ. Hector Pasten told Scientific American that the proof combines two of his own ideas in a way he had not anticipated. Zhi-Wei Sun, who had reduced the integer problem to few unknowns, took the result as given and brought the rational version down to 86 unknowns. On Calegari’s blog a commenter relayed a complaint that the paper miscites earlier work on subrings of ℚ; another commenter doubted that this mattered. No line-by-line check has been published.
Sources
8 sources from 5 sites. Numbers match the chips in the text.
8 sources: 4 primary, 1 press, 3 reactions
Primary
- OpenAI: Hilbert’s tenth problem over the rational numbers (6 Oct version)github.com, paper
- OpenAI: A pointwise 2-converse for elliptic curves with rational two-torsiongithub.com, paper
- OpenAI: Fontaine–Mazur modularity at the prime 2 (input to the proof)github.com, paper
- arXiv 2610.20691: Zhi-Wei Sun, 86 unknowns for Hilbert’s tenth problem over the rationalsarxiv.org, paper
Press
- Scientific American: The most exciting claims from OpenAI’s heap of new proofs (Pasten on Hilbert’s tenth)scientificamerican.com, press
Reactions
- Persiflage (Frank Calegari): A Brief Digest of Some New OpenAI Results (8 Oct)galoisrepresentations.org, discussion
- Bill Gasarch: Conjecture that I thought would not be solved in my lifetime....were (11 Oct)blog.computationalcomplexity.org, discussion
- oai-math-revised: community notes on the Hilbert’s tenth manuscriptgithub.com, discussion
Changes
- Filed (families 001–125; manuscript introduction and references, Persiflage digest and comments, SciAm, Gasarch and Sun’s arXiv abstract read)