Hilbert’s tenth problem over ℚ
No algorithm can decide whether a whole-number polynomial equation has a solution in fractions, the paper claims.
Hilbert asked in 1900 for a method to decide whether a polynomial equation has whole-number solutions. In 1970 that was shown to be impossible. The same question for fractions, the rational numbers, stayed open. This paper claims that it is impossible too. No computer program can sort every such equation correctly into solvable and unsolvable over the fractions.
The explanation
Take the equation x² + y² = 2. It has the solution x = 1, y = 1. It also has the fraction solution x = 1/5, y = 7/5, since 1/25 + 49/25 = 50/25 = 2. Now take x² = 2. No fraction works, because the square root of 2 is irrational. Hilbert's tenth problem over the rationals asks for one method that sorts every such equation. The method must say whether some choice of fractions makes it true.
The equations allowed are polynomials with whole-number coefficients. The number of unknowns can be anything, and it is part of the input. Fractions are the rational numbers, written a/b with whole numbers a and b, and b not zero.
Hilbert first asked the question in 1900 for whole-number solutions. In 1961, Davis, Putnam and Robinson did most of the work, with an extra power operation allowed. In 1970, Matiyasevich supplied the last step. Their combined theorem says whole-number equations can encode any condition whose cases a program can list. So no algorithm decides whether a whole-number equation has a whole-number solution.
For fractions, the question stayed open. A search through all fractions will find a solution if one exists. If none exists, the search runs forever and never says so. The natural plan was to describe the whole numbers inside the fractions using only equations. Then the old answer would carry over. In 1949, Julia Robinson described them with a formula that uses both "for every" and "there is". Poonen in 2009 and Koenigsmann in 2016 gave simpler formulas of that mixed kind. None gave the equation-only description the plan needs. A 1995 conjecture of Mazur, if true, would rule that description out.
Nearby cases were settled. In 2003, Poonen proved the problem undecidable for some rings of fractions whose denominators use almost all primes. Recent work by Koymans and Pagano, and separately by Alpöge, Bhargava, Ho and Shnidman, settled rings of integers of number fields. Those rings are different from the field of all fractions.
The paper claims the full answer is no. No algorithm decides whether a whole-number polynomial has a rational zeroSource 1. It also claims the problem is exactly as hard as the halting problem, the question of whether a program ever stopsSource 2. That stays true for polynomials of degree at most fourSource 3.
The approach avoids describing the whole numbers directly. For each polynomial f, it builds an endless list of finite tests. Each test asks finitely many questions of the form "does this equation have a fraction solution?". The paper claims f has a whole-number zero exactly when every test passes. Suppose a method for fractions existed. Then two searches could run side by side. One hunts for a whole-number zero, and the other hunts for a failed testSource 4. One of them must stop.
The hard part is showing a failed test exists whenever f has no whole-number zeroSource 5. The tests attach to each number a point on an elliptic curveSource 6. That is a cubic curve whose points can be added like numbers. Each attached point is a multiple of one fixed point, so it carries a whole-number index. The proof measures size by height, roughly the number of digits in a fraction. If every test passed, a logic tool called compactness would give an imagined zero, perhaps with infinitely large entries. Height bounds at carefully chosen primes cap the indices of its entries. That leaves only ordinary whole numbers, and f would have a real whole-number zero.
The proof leans on two other papers in the openai/math release. One is a 2-converse for elliptic curves with a point of order 2Source 7. It is the second paper of this resultSource 8. The other is a modularity theorem at the prime 2, claimed in a separate paperSource 9. Neither has a published specialist review.
How sure is this? No Lean formalization is listed for these papers in the openai/math release. So no computer proof checker has checked them. No specialist review has been published.
What it doesn't show. This does not give a method for any special family of equations, such as curves of one fixed kind.
Where it came from
This result comes from an unreleased OpenAI model, published in the openai/math release on 6 October 2026.
Hype check
Landmark only if this paper and the claimed 2-converse and modularity results it uses all hold. No Lean formalization is listed and no specialist review is published.
Words used
- Rational number
- A fraction a/b of whole numbers, with b not zero.
- Hilbert's tenth problem
- Hilbert's 1900 request for a method that decides whether a polynomial equation has whole-number solutions.
- Algorithm
- A step-by-step procedure a computer can follow that always finishes with an answer.
- Halting problem
- The question of whether a given computer program ever stops. No algorithm answers it for every program.
- Elliptic curve
- A cubic curve, such as y² = x³ + ax + b, whose points can be added like numbers.
- Height
- A size measure for a fraction, roughly the logarithm of its larger numerator or denominator.
- Compactness
- A logic principle: if every finite batch of conditions can be met, some possibly unusual structure meets them all.
The original papers (2)
All quoted lines (9)
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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.
Hilbert’s tenth problem over the rational numbers Back to text -
Then has Turing degree , the degree of the halting problem.
Hilbert’s tenth problem over the rational numbers Back to text -
rational solvability for integral polynomials of total degree at most four;
Hilbert’s tenth problem over the rational numbers Back to text -
If a rational-solvability algorithm existed, we could run two searches in parallel: enumeration of integer zeros, and enumeration of tests until one fails.
Hilbert’s tenth problem over the rational numbers Back to text -
The arithmetic work is to prove that a failed test exists whenever has no integer zero.
Hilbert’s tenth problem over the rational numbers Back to text -
The tests attach an elliptic point to each , compatibly with addition, and require to be nonidentity when .
Hilbert’s tenth problem over the rational numbers Back to text -
Section … proves the pole-parity condition; its passage from two-descent data to rational points uses the pointwise -converse and the Cassels–Tate pairing.
Hilbert’s tenth problem over the rational numbers Back to text -
Here we give a separate full treatment of the rational-two-torsion case, developing its integral interpolation and finite twist constructions within this paper; the broader converse is not an input to the proof.
A pointwise 2-converse for elliptic curves with rational two-torsion Back to text -
Its modularity step uses the odd regular Fontaine–Mazur theorem at , whose statement has no hypothesis on the residual image.
Hilbert’s tenth problem over the rational numbers Back to text
Other sources
- OpenAI, “Hilbert's tenth problem over the rational numbers”, openai/math release preprint (October 6, 2026)
Enumerating rational tuples will eventually find any rational zero, but gives no answer for an equation without one.
- OpenAI, “Fontaine–Mazur modularity at the prime 2”, openai/math release preprint (October 6, 2026)
- openai/math release notes (2026): model provenance and review status
manuscripts and supporting proof artifacts produced by an internal OpenAI model
- J. Robinson, “Definability and decision problems in arithmetic”, Journal of Symbolic Logic 14 (1949), 98–114
- M. Davis, H. Putnam and J. Robinson, “The decision problem for exponential Diophantine equations”, Annals of Mathematics 74 (1961), 425–436
- M. Davis, “Hilbert's tenth problem is unsolvable”, American Mathematical Monthly 80 (1973), 233–269
- B. Poonen, “Characterizing integers among rational numbers with a universal-existential formula”, American Journal of Mathematics 131 (2009), 675–682
- J. Koenigsmann, “Defining ℤ in ℚ”, Annals of Mathematics 183 (2016), 73–93
- B. Poonen, “Hilbert's tenth problem and Mazur's conjecture for large subrings of ℚ”, Journal of the AMS 16 (2003), 981–990
- P. Koymans and C. Pagano, “Hilbert's tenth problem via additive combinatorics”, Journal of the AMS (2026)