One great problem, three kinds of immersion
From a story to your own conjecture
Problem stories open curiosity. The problem workshop is the next place, where curiosity becomes a testable idea.
- 1 · DiscoverYou are hereMeet the questionLearn when it appeared and why it still holds people’s attention.
- 2 · ChallengeTest one caseMake your first observation in five minutes with a drawing, calculation, or colors.
- 3 · DevelopBuild on ideasPublish an observation and grow it through comparisons, conjectures, and counterexamples.
Hilbert’s Tenth over ℚ — Can Every Equation Be Decided?
The equation 2x=1 has no integer solution but has a rational one. Allowing fractions changes the question of whether a single method can decide solvability for every polynomial.
Small experiments are possible, but a complete proof sits at the frontier of modern mathematics.
The starting level measures how easily you can understand and test small cases. It is not the difficulty of a complete proof.
Jump to your first five minutesAI Frontier 2026
AI Frontier 2026 — a claim is not yet a theorem
Below are claims that AI labs have published about this problem. Read what is claimed together with what is not.
- OpenAIAnnounced 2026-10-06openai/math family 004Same problem
Hilbert’s tenth over ℚ: a claimed absence of a rational-solvability algorithm
No public formalizationCommunity review- Claim
- Claims that no always-terminating algorithm decides rational solvability of integer-coefficient polynomials, with the variable count part of the input. This is the main theorem of the October 6 manuscript; Sun’s follow-up claims the 86-variable case.
- Not claimed
- Do not conflate this with the 1970 result over ℤ or the full first-order theory of ℚ. The family 004 listing checked on October 28 has no Lean link. Sun’s paper uses height and Selmer inputs from the original; its existence does not establish independent verification of the whole proof or completed peer review.
- MathVoyage's call
- The new rational-version page starts at open and marks the 2026 resolution claim as under review. This editorial boundary separates an announcement from established acceptance; it is not a verdict that the proof is false.
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No problem status on this site has changed on the strength of a claim alone.
See the puzzle visually
Finding a solution ≠ a decision that terminates on every input
Separate the 1970 theorem over ℤ from the 2026 claim over ℚ under review. The year 1900 refers to the original integer problem.
Problem statement
Is there an algorithm that takes an integer-coefficient polynomial f(x₁,…,xₙ) and always terminates with a yes/no answer about the existence of a rational zero? The number of variables n is part of the input. The question concerns a terminating decision procedure for all inputs, not one equation or merely a fast algorithm.
The story of this puzzle
Hilbert’s original 1900 problem concerned integer solutions. Matiyasevich completed the work of Davis, Putnam, and Robinson in 1970, establishing that no general integer-solution decision algorithm exists. The conclusion does not transfer automatically to rational unknowns. Enumerating rational tuples eventually finds a solution when one exists, but deciding when to stop if none exists is a separate issue.
The OpenAI family 004 manuscript released on October 6, 2026 claims a negative resolution over ℚ as well. Its main theorem allows the number of variables to be part of the input. The public family listing checked on October 28 has no Lean link for this result. That is neither a disproof nor evidence that no formalization exists anywhere.
Zhi-Wei Sun’s follow-up preprint, submitted October 14 and revised October 19, uses height and Selmer-theoretic inputs from the manuscript to claim undecidability with 86 variables. A human follow-up is meaningful, but is not independent verification of the entire original proof or completed peer review. This page therefore pairs an initial open status with the claim under review. Undecidability of the full first-order theory of ℚ is also distinct from the purely existential question here.
Try it yourself
Compare rational solutions of 2x=1 and x²=2. The first has x=1/2; the second has no rational root. Does deciding these individual examples guarantee a single algorithm for every multivariable polynomial? When can an ongoing search legitimately answer “none”?
Beyond MathVoyage
Curated sources and problems. Bring one discovery back from OEIS, Project Euler, MathOverflow, or arXiv.