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The Millennium Prize Problems
Seven problems, a million dollars each, and one clear lesson for anyone learning this field: the hard part of a hard problem is almost never the arithmetic.
In 2000 the Clay Mathematics Institute named seven problems and attached $1,000,000 to each. One is solved: Grigori Perelman proved the Poincaré conjecture and then declined the money, and the other six are open.
Difficulty alone did not decide the choice. Each one is load-bearing, because a whole field has been built on top of an assumption nobody can prove, and that earns them a prize.
One of them is live news as this page is written. On 8 September 2026 OpenAI announced a machine-generated, formally verified proof of a blow-up result for Navier–Stokes. The Clay Institute has not accepted it, credit is being disputed, and section 05 walks through exactly what was and was not proved.
01Why a prize, and why these seven
The list was designed to be a century's worth of work, in the tradition of a much older list.
In 1900 David Hilbert stood up in Paris and named 23 problems he thought should shape the coming century. Work on that list produced much of twentieth-century mathematics, and several of the problems turned out to be unanswerable in the form he posed them.
The Clay Mathematics Institute made the parallel explicit. It announced its seven problems in Paris, in May 2000, exactly a hundred years later. The prize money is real but it is not the point. The list is a statement about where the gaps are.
Two features make a problem prize-worthy and not merely unsolved.
- Something large already depends on the answer. Cryptography assumes P ≠ NP. Analytic number theory routinely publishes results "assuming the Riemann Hypothesis". Physicists compute with a quantum field theory nobody has constructed rigorously. These fields are not waiting politely; they have moved in and built.
- The obvious approaches are known to fail. For the deepest problems on the list there are theorems saying that whole classes of proof technique cannot work. This is a much stronger statement than "we tried and got stuck".
You cannot win by posting a proof. The Clay rules require publication in a peer-reviewed journal of worldwide repute, followed by roughly two years during which the proof is generally accepted by the mathematical community.
That gap between correct and accepted is not bureaucracy. Wiles's first proof of Fermat's Last Theorem had a hole in it that took a year to fix. Section 05 is about what happens when a machine produces a proof faster than the community can absorb it.
02The seven, in one line each
Read the table first, since the sections after it expand the three that a machine-learning reader has the most reason to care about.
| Problem | Field | What it asks | Status |
|---|---|---|---|
| P vs NP | Computational complexity | If a solution can be checked quickly, can it be found quickly? | Open |
| Riemann Hypothesis | Number theory | Do all non-trivial zeros of the zeta function lie on one vertical line? | Open |
| Navier–Stokes | Fluid dynamics / PDE | Do the equations of fluid flow always have smooth solutions, or can they break? | Open · disputed claim, Sept 2026 |
| Yang–Mills mass gap | Quantum field theory | Can the theory behind the strong force be built rigorously, with a mass gap? | Open |
| Hodge conjecture | Algebraic geometry | Are certain topological features of a shape always traceable to algebraic equations? | Open |
| Birch–Swinnerton-Dyer | Number theory | Does an elliptic curve's L-function encode how many rational points it has? | Open (special cases proved) |
| Poincaré conjecture | Topology | Is every simply connected closed 3-manifold a sphere? | Solved — Perelman, 2002–03 |
The one that is finished is worth a paragraph of its own. Grigori Perelman posted three preprints to arXiv in 2002 and 2003, completing a programme Richard Hamilton had started with Ricci flow: deform a shape according to its own curvature and watch what it becomes. He never published in a journal, declined the Fields Medal in 2006, and declined the million dollars in 2010.
03P vs NP, and why it is in your loss function
This is the problem on the list that touches machine learning directly, every day, whether or not anyone says so.
Some problems are easy to check and appear hard to solve. Given a completed sudoku you can verify it in seconds. Given a blank one you cannot. P is the class of problems solvable in polynomial time. NP is the class whose solutions can be verified in polynomial time. The question is whether those two classes are the same.
Almost everyone believes P ≠ NP. Nobody can prove it, and three separate results explain why the obvious attacks fail, which makes this more than a gap in the literature.
- Relativization (Baker, Gill and Solovay, 1975). Any proof that works by simulating machines with access to an oracle cannot settle the question, because there are oracles making P = NP and others making P ≠ NP.
- Natural proofs (Razborov and Rudich, 1994). This result shows that a large family of circuit-complexity arguments, the kind that had produced the field's best lower bounds, provably cannot separate P from NP unless strong cryptography is impossible.
- Algebrization (Aaronson and Wigderson, 2008). The technique that got around relativization runs into its own barrier, and this result shows it.
Training a neural network to its global optimum is NP-hard in general. That has been known since Blum and Rivest showed it for a network with three nodes in 1992. Non-convex optimisation is not a temporary embarrassment waiting for a smarter algorithm.
So the field does something specific and defensible: it stops asking for the optimum. Gradient descent finds a good-enough local solution, and the whole practice of machine learning is built on the observation that good-enough generalises well. Complexity theory shows that this is a sound trade and is not a way of giving up.
The same logic runs through the rest of the stack. Exact inference in a general graphical model is intractable, so we sample. Optimal decoding over all possible sequences is intractable, so we use beam search and accept that it is greedy. Many of the heuristics you have learned exist because somebody proved the exact version is out of reach.
04The Riemann Hypothesis: a load-bearing conjecture
The clearest example on the list of a field that decided not to wait.
The Riemann zeta function $\zeta(s)$ has zeros. Some are boring and easy to locate. The rest, the non-trivial zeros, are conjectured to lie exactly on the vertical line where the real part of $s$ is one half:
Stated in 1859, in an eight-page paper, almost in passing.
This sounds like a technicality about one function, but the zeros of $\zeta$ control how irregularly the prime numbers are distributed. The Prime Number Theorem tells you roughly how many primes lie below a given size. The Riemann Hypothesis would tell you how far off "roughly" can be, and it would be the tightest possible answer.
Two facts about its status are worth holding together.
- The numerical evidence is overwhelming. Something on the order of ten trillion zeros have been checked. Every one is on the line.
- The numerical evidence proves nothing. Ten trillion is not a proof, and mathematicians have been burned before by patterns that hold for enormous ranges and then fail. An analogous statement over finite fields, the Weil conjectures, has been proved, which is encouraging and not the same thing.
Meanwhile hundreds of published theorems begin with the phrase "assume the Riemann Hypothesis". An entire literature is conditional on an unproved statement. If it fails, that literature does not merely lose a tool; parts of it become false.
06The other three, briefly
Less relevant to this site's subject matter, and worth a mention.
The strong force is described by a quantum field theory that physicists compute with constantly. The problem is to construct that theory rigorously on four-dimensional space, and prove it has a mass gap: a strictly positive lowest excitation energy.
Why it matters: the mass gap is why the strong force has short range even though gluons are massless. Lattice simulations show it clearly. Nobody has built the theory they are simulating.
On a smooth projective complex variety, certain classes identified by topology are conjectured to always come from actual algebraic subvarieties, up to rational coefficients.
Why it matters: it is a bridge between topology and algebra. Proving it would say that a shape's holes are always cut out by equations, which is the kind of statement that reorganises a field.
For an elliptic curve over the rationals, the conjecture says the rank of its group of rational points equals the order of vanishing of its L-function at $s=1$.
Why it matters: it connects counting solutions to an analytic object. Special cases are proved, notably for rank 0 and 1 under conditions, via work of Gross, Zagier and Kolyvagin. The general case is open.
07Where AI touches this
The useful question is not whether machines will solve these. It is which kind of mathematical work they are currently good at, because the answer is specific and it generalises far beyond mathematics.
The last three years produced a real track record, none of it hand-waving:
- Formal infrastructure. Lean and its library mathlib let a proof be machine-checked line by line. Everything else here depends on this substrate, and humans built it.
- Competition mathematics. DeepMind's AlphaProof, trained with reinforcement learning over auto-formalised problems, reached silver-medal level at the 2024 International Mathematical Olympiad. In 2025 both DeepMind and OpenAI reported gold-medal-level performance, five problems out of six, using general-purpose reasoning models.
- Finding constructions. FunSearch improved a bound on the cap set problem. AlphaEvolve found a way to multiply two 4×4 matrices with 48 scalar multiplications, beating the 49 that Strassen's method had held since 1969.
- Chipping at open problems. An AI agent resolved a small number of formalised open Erdős problems, on the order of nine out of several hundred attempted, with human experts checking that the formal statements matched the originals.
Look at what those wins have in common. A 48-multiplication scheme is hard to find and trivial to check. A Lean proof is hard to construct and mechanically checkable. An IMO problem has an answer you can grade.
This is the shape of problem current AI is strong on: enormous search space, cheap and reliable verification. Where a verifier exists, you can let a model search badly and often, and throw away everything that fails.
Now look at what is missing from the list. Nobody is attacking the Riemann Hypothesis, Hodge or Birch–Swinnerton-Dyer this way, because those do not appear to need a search through candidate objects. They appear to need a new idea about what the right objects are. There is no verifier for that. Page 37 shows the same dividing line running through every industry.
08What breaks
Mostly in how these results get reported, which is worth being able to see through.
- "AI solved a Millennium Problem" collapses three different claims. A proof existing, a proof being verified, and a proof being accepted as resolving the prize problem are separate events, sometimes years apart.
- Formally verified is not the same as relevant. Lean guarantees the argument is sound. Whether the formal statement is the intended statement is a human judgement, and that is precisely where the current dispute sits.
- The machine's contribution is easy to overstate. The Navier–Stokes attack rests on analytic techniques developed by named humans over years. The model did a hard, specific piece of work on top of that foundation.
- Numerical evidence is not proof, and this bites in ML too. Ten trillion zeros on the line proves nothing about the next one. A benchmark passing on ten thousand examples proves nothing about the distribution you have not sampled.
- "Unsolved" does not mean "untouched". Every problem here has a large partial literature. Special cases, analogues and reformulations are where the actual progress lives.
- Prizes distort attention. Plenty of consequential open problems carry no million dollars. The list is a snapshot of what seemed important in 2000. It is not a ranking of importance today.
09Interview questions
BeginnerWhat does P vs NP ask, and why does it matter for machine learning?
P is the class of problems solvable in polynomial time; NP is the class whose candidate solutions can be verified in polynomial time. The question is whether the two classes coincide, and the near-universal expectation is that they do not. It matters for machine learning because the exact versions of many things we do are provably intractable: finding the global optimum of a neural network's loss is NP-hard, exact inference in general graphical models is intractable, and optimal sequence decoding is infeasible. That is the justification for the entire toolkit of approximations we actually use, from gradient descent to sampling to beam search. Complexity theory is what tells you those are principled trades rather than shortcuts.
BeginnerWhich Millennium Prize Problem has been solved, and by whom?
The Poincaré conjecture, which asks whether every simply connected closed three-dimensional manifold is homeomorphic to the three-sphere. Grigori Perelman proved it in a series of arXiv preprints in 2002 and 2003, completing a programme based on Richard Hamilton's Ricci flow, which deforms a manifold according to its own curvature. He never submitted the work to a journal, declined the Fields Medal in 2006 and declined the million-dollar prize in 2010. The other six problems remain open, with a disputed claim on Navier–Stokes as of September 2026.
IntermediateWhy do complexity-theoretic barriers make P vs NP different from an ordinary open problem?
Because there are theorems ruling out entire families of proof technique. Relativization, from Baker, Gill and Solovay in 1975, shows that any argument which still works when machines are given an oracle cannot resolve the question, since some oracles make the classes equal and others separate them. Natural proofs, from Razborov and Rudich in 1994, shows that a broad class of circuit lower-bound arguments cannot separate P from NP unless strong pseudorandom functions fail to exist. Algebrization, from Aaronson and Wigderson in 2008, shows the main technique that evaded relativization has its own limit. Together they mean progress requires a genuinely new kind of argument, which is a much stronger statement than nobody having found the proof yet.
IntermediateWhat does it mean for the Navier–Stokes equations to "blow up", and why would that matter?
Blow-up means a solution that starts smooth with finite energy develops a singularity in finite time, with velocity becoming unbounded at some point. It matters because it would show the equations are an incomplete model of a real fluid: water does not reach infinite speed, so a blow-up marks the point where the mathematics stops describing physics. It also matters for the prize specifically, because Fefferman's official statement offers the award for either direction. Two of the four statements ask for global smoothness with no external force, and the other two ask for a breakdown example, where a smooth forcing term satisfying stated conditions is permitted.
IntermediateA model produces a Lean-verified proof of an open conjecture. What can and cannot be concluded?
You can conclude that the formal statement follows from the axioms with no gaps, which is a stronger correctness guarantee than a human referee provides. What you cannot conclude is that the formalised statement is the conjecture people care about. Translating an informal problem into a formal one involves choices about definitions, quantifiers and side conditions, and a faithful translation is a human judgement rather than something the proof checker validates. So the remaining work after verification is a mathematical and social process: experts confirm the statement is the intended one, and the community absorbs the argument. This is exactly the gap at issue in the September 2026 Navier–Stokes claim.
DeepWhat kinds of mathematical problem is current AI good at, and why?
Problems with an enormous search space and a cheap, reliable verifier. Finding a scheme that multiplies 4×4 matrices in 48 multiplications is hard to discover and trivial to check, which is why AlphaEvolve could improve on a bound that had stood since 1969. A Lean proof is hard to construct and mechanically checkable. Competition problems have gradeable answers. In all these cases a model can search unreliably and at volume, discarding everything that fails verification. The problems where this does not apply are the ones needing a new conceptual framework rather than a search over candidate objects, which is the situation for the Riemann Hypothesis, Hodge and Birch–Swinnerton-Dyer. There is no verifier for "is this the right definition", so there is nothing to search against.
DeepHundreds of papers assume the Riemann Hypothesis. Is that sound practice?
It is standard and it is explicitly flagged, which is what makes it defensible. A conditional theorem is a real result: it establishes an implication, and it maps out what would follow if the conjecture holds. The evidence base is also unusually strong, with on the order of ten trillion zeros verified on the critical line and a proved analogue over finite fields. The risk is nonetheless real and asymmetric. If the hypothesis fails, conditional results do not merely lose a convenience; some of them become false, and the error terms that depend on it are exactly the quantitative content of the work. The practice is sound because the dependency is declared, not because the assumption is safe.
10Go deeper
Start with primary sources. On the September 2026 story especially, read the originals and skip the headlines.