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Millennium Prize Problems and AI
What has been solved, what has been claimed, and what remains in mathematics’ seven hardest problems.
The Millennium Prize Problems are seven questions selected by the Clay Mathematics Institute as some of the deepest challenges in modern mathematics. Each carries a US$1 million prize.
One problem, the Poincare Conjecture, has been solved. Five remain officially unsolved. The seventh, Navier-Stokes, is now at the centre of a major AI-generated solution claim.
The status at a glance
1 problem
The Poincare Conjecture was solved by Grigori Perelman. The original solution was produced by a human, not AI.
1 AI claim
OpenAI has published a proposed Navier-Stokes solution and a Lean formalization. The Clay process is not complete.
5 problems
Birch and Swinnerton-Dyer, Hodge, P versus NP, Riemann, and Yang-Mills remain officially unsolved.
| Problem | Clay status | AI state | Kingy label |
|---|---|---|---|
| Poincare Conjecture | Solved | The original solution was produced by Grigori Perelman. | Solved |
| Navier-Stokes | Active | OpenAI has published a proposed solution and Lean formalization. | AI claim |
| Birch and Swinnerton-Dyer | Unsolved | No authoritative completed AI solution identified. | Open |
| Hodge Conjecture | Unsolved | No authoritative completed AI solution identified. | Open |
| P versus NP | Unsolved | No authoritative completed AI solution identified. | Open |
| Riemann Hypothesis | Unsolved | AI has produced progress on a related zero-density result, not a solution. | Related progress |
| Yang-Mills and Mass Gap | Unsolved | No authoritative completed AI solution identified. | Open |
Clay’s official overview lists the Poincare Conjecture as solved and the Birch and Swinnerton-Dyer, Hodge, P versus NP, Riemann, and Yang-Mills problems as unsolved. Its Navier-Stokes page currently uses the status “Active.” Clay Mathematics Institute: The Millennium Prize Problems.
What counts as an official solution?
“AI solved a Millennium Prize Problem” can describe several different things. Kingy uses this evidence ladder:
- 1. Model claimAn AI lab says a system found a proof or disproof.
- 2. Public argumentThe reasoning is available in a paper, preprint, or technical write-up.
- 3. FormalizationSome or all of the argument is encoded in Lean or another proof assistant.
- 4. Independent checkingResearchers outside the originating team inspect the mathematics and scope.
- 5. Community acceptanceThe result survives publication, criticism, correction, and broader review.
- 6. Clay recognitionThe Clay Institute determines that its prize conditions have been met.
These are milestones, not synonyms. A Lean file can show that a formally stated theorem follows from formally stated premises. Humans still need to check that the formal theorem is the same theorem the researchers intended to prove and that it addresses the official problem rather than a nearby variant.
Clay’s rules require publication in a qualifying outlet, two years of publication, and general acceptance in the global mathematics community before the Institute will consider awarding the prize. Read Clay’s rules.
Navier-Stokes: the first major AI solution claim
Can smooth fluid motion break down?
The Navier-Stokes equations describe how fluids such as water and air move. The mathematical problem asks whether smooth, three-dimensional, incompressible fluid motion can remain smooth for all time, or whether a solution can develop a singularity in finite time.
The difficulty is proving what nonlinear interactions must do over an unlimited period of time.
What OpenAI claims
On September 8, 2026, OpenAI announced that an internal system had produced an analytical proof showing that a smooth, forced Navier-Stokes flow can develop a finite-time singularity while retaining finite energy. OpenAI says the result establishes statements C and D in the official Clay formulation.
The company reports that roughly 10,000 coordinating agents worked on the problem, generating about 2.7 million internal messages. It says the proof was found after approximately 88 hours and that a further 17 hours produced a Lean formalization. Read OpenAI’s announcement. OpenAI updated the same page on September 10 with the provenance investigation described here.
What has been established?
- OpenAI has disclosed a specific proposed proof.
- The company has released a Lean formalization.
- The result addresses the official formulation rather than only a numerical simulation.
- OpenAI says it does not intend to claim the US$1 million Clay prize.
- Clay still lists Navier-Stokes as “Active.”
The related Euler result
The story is connected to separate work by Tristan Buckmaster and Levent Alpoge at Anthropic on a forced version of the three-dimensional Euler equations. Euler equations remove the viscosity term found in Navier-Stokes, so the problems are related but not identical. OpenAI’s September 10 update says its Euler result was unforced, while the Alpöge-Buckmaster result used external forcing.
A result for forced Euler is not automatically a result for the Clay Navier-Stokes problem. Quanta Magazine’s report explains the technical relationship and the priority dispute. Nature’s report describes the claim as a company announcement that still requires scrutiny.
The five problems that remain officially open
Birch and Swinnerton-Dyer
This conjecture concerns elliptic curves and predicts a relationship between rational solutions and an associated L-function. AI could help search curve families, spot patterns, suggest intermediate conjectures, and formalize algebraic arguments. Numerical evidence would not prove the universal statement. Clay overview.
Hodge Conjecture
The Hodge Conjecture asks how much of the shape of an object defined by polynomial equations can be described using algebraic subobjects. AI could help search examples, classify special cases, and assist with formal proof development. Clay overview.
P versus NP
P versus NP asks whether every problem whose answer can be checked efficiently can also be solved efficiently. AI can help investigate proof complexity and lower-bound techniques, but solving difficult instances would not establish P = NP. Clay overview.
Riemann Hypothesis
The hypothesis concerns the non-trivial zeros of the Riemann zeta function. Anthropic reports that Claude improved a related lower bound from 41.6% to 67.2%, while saying Claude did not solve the hypothesis. Anthropic’s report · Clay overview.
Yang-Mills and the Mass Gap
The problem asks for a rigorous construction of a quantum Yang-Mills theory and a proof of a positive mass gap. AI could assist with symbolic calculations, lattice-gauge experiments, literature mapping, and formalization. Clay overview.
The one problem already solved: Poincare
In 1904, Henri Poincare asked whether the three-dimensional sphere is characterized as the unique simply connected three-manifold. Grigori Perelman solved the problem through work on Ricci flow, posting the key preprints in 2002 and 2003. Clay lists the problem as solved.
What AI is contributing to mathematics
Formal proof and proof assistants
Lean and similar systems let researchers express definitions, theorems, and proof steps in a machine-checkable language. Google DeepMind’s AlphaProof contributed to a silver-medal-level result on the 2024 International Mathematical Olympiad. That is evidence of progress in formal reasoning, but an Olympiad problem is not a Millennium Prize Problem. Google DeepMind’s report.
Multi-agent research workflows
The Navier-Stokes announcement is an example of a workflow built around many coordinating agents. Agents can divide a problem into subproblems, compare approaches, call software tools, and ask other agents to critique or formalize an argument. More agents can generate more ideas, but they can also generate more duplicated or incorrect reasoning.
Human-AI collaboration
OpenAI’s First Proof experiment had AI systems attempt research-level problems while mathematicians inspected the arguments. OpenAI reported that at least five of ten attempts had a high chance of being correct, while acknowledging that some remained under review and one initially promising attempt was later judged incorrect. OpenAI’s report.
AI claims tracker
Updated as new papers, formalizations, reviews, and corrections appear.
| Date | Team | Target | Claim or result | Verdict |
|---|---|---|---|---|
| Sep. 8, 2026 | OpenAI | Navier-Stokes | Proposed finite-time singularity result with Lean formalization. | Proposed AI solution |
| Sep. 10, 2026 | OpenAI | Navier-Stokes provenance | Published investigation update: prior Buckmaster Codex prompts could not have influenced the result; OpenAI also clarified the Euler forcing distinction. | Evidence update |
| Sep. 2026 | Buckmaster and Alpoge | Forced 3D Euler | AI-assisted work on a related fluid-equation problem. | Related result |
| Aug. 2026 | Anthropic | Riemann-adjacent result | Claude reportedly improved a lower bound from 41.6% to 67.2%. | Related progress |
| Feb. 2026 | OpenAI | First Proof | Several research-level proof attempts. | Capability evidence |
| Jul. 2024 | Google DeepMind | IMO mathematics | AlphaProof and AlphaGeometry solved four of six IMO problems at silver-medal level. | Formal milestone |
Frequently asked questions
Has AI officially solved a Millennium Prize Problem?
Not under Clay’s prize process. OpenAI has published a proposed Navier-Stokes solution and a Lean formalization, but Clay still lists the problem as active. OpenAI’s September 10 provenance update does not change that status.
Which problems remain unsolved?
Birch and Swinnerton-Dyer, Hodge, P versus NP, Riemann, and Yang-Mills are officially unsolved. Navier-Stokes is active while the new claim is evaluated. Poincare is the one officially solved problem.
Does a Lean proof automatically settle the mathematics?
No. Lean verifies a formally stated theorem from formally stated premises. Researchers must establish that the formal statement matches the intended claim and the official problem.
Is a related problem the same as a Millennium Prize Problem?
No. A result about Euler is not automatically a result about Navier-Stokes, and a better zeta-function bound is not a proof of the Riemann Hypothesis.
Can AI prove the Riemann Hypothesis or P versus NP?
There is currently no authoritative AI solution to either problem. Claims should be judged from the exact theorem, full proof, formal artifacts, independent review, and community acceptance.
Sources and official references
- Clay: The Millennium Prize Problems
- Clay: Rules for the Millennium Prize Problems
- Clay: Navier-Stokes Equation
- OpenAI: Navier-Stokes announcement and September 10 update
- Quanta Magazine: AI Has Solved One of Math’s $1 Million Millennium Prize Problems
- Nature: OpenAI claims huge maths breakthrough on a famed Millennium Problem
- Anthropic: Learning more about Claude’s mathematical capabilities
- OpenAI: Our First Proof submissions
- Google DeepMind: AlphaProof and AlphaGeometry
Page maintained by Kingy AI. Last verified September 11, 2026 at 4:40 PM PDT.