Science1 publisher2 min readPublished
Scientific American puts Birch and Swinnerton-Dyer first in line to fall to AI
With Navier-Stokes claimed and five Millennium problems left, the ordering of what falls next turns on which conjectures a single counterexample would settle. Five million dollars of Clay prize money is still unclaimed.
The Scientist · Science desk
What happened
- OpenAI said on September 8, 2026 that an artificial intelligence model may have solved the Navier-Stokes problem, one of the seven Millennium Prize Problems the Clay Institute of Mathematics selected in 2000.
- That takes the tally of solved Millennium problems to two, the other being Grigori Perelman's 2003 proof of the Poincare conjecture.
- OpenAI told the New York Times it has already made significant progress on another of the problems, without saying which one.
- Scientific American introduced the five remaining problems in order of how likely each is to be resolved by the technology, an ordering it attributes to most experts.
- A year ago the reference point for these models was the Mathematics Olympiad, which they were only barely managing, and many researchers doubted the models were useful in research at all.
Compiled by The ScientistSomething wrong?How this is made
Why it matters
- contradiction The public ranking of what falls next is being assembled from outside the labs while OpenAI already says it holds significant progress on one of the five and has not said which, so the speculation may be forecasting a result someone has in hand.
- capability Because the prize admits a counterexample as a solution, a lab can win $1 million by handing over one object that other mathematicians can test, without producing a theory.
- constraint If the opening is the under-searched refutation side of conjectures people believe, then compute goes first at problems that can be killed by an example, and the ones needing structural proof stay where they were.
- precedent A claimed solution announced by a company, with no verifier named in the account, sets the terms on which the next five will arrive.
Birch and Swinnerton-Dyer's conjecture dates to the early 1960s and concerns elliptic curves, the family of equations y^2 = x^3 + ax + b [10][11]. Those curves are central to cryptography and were the key to solving Fermat's last theorem [20]. The hard quantity is the rank: the number of independent rational starting points on a curve, from which further rational points follow by simple arithmetic [12]. In a computer-aided search, Birch and Swinnerton-Dyer found evidence of a criterion that answers the rank question quickly, and whether it gives the right rank for every curve has stayed open since [13]. One route to the million dollars is a single curve that fails the criterion [14][3].
Scientific American's stated reason for expecting a model to find such a curve has to do with where human effort has gone: "people often invest more resources in proving a hypothesis they believe to be true than in a possible refutation," the magazine wrote [15]. Language models have been good at producing examples of this kind, the article says [16].
The article does not say who has checked the Navier-Stokes argument or whether it has been refereed; it reports that the mathematics community is still in shock [19][18]. The wording of the claim also moves inside the same piece. OpenAI announced that a model may have solved the problem [1], and the article then counts Navier-Stokes among the two Millennium problems solved to date, next to Grigori Perelman [4].
Perelman proved the Poincare conjecture in 2003, twenty-three years before the September announcement [4][1][2]. Each of the five remaining problems carries a $1-million prize, which leaves $5 million of Clay Institute money outstanding [3][5][1].
Second in the ordering is the Hodge conjecture, which William Vallance Douglas Hodge put forward in 1950 and which the article calls arguably the most abstract of the seven [9].
Checking the two kinds of claim is not the same work. A candidate counterexample to Birch and Swinnerton-Dyer can be tested, one curve against one criterion [13][14]. A claimed proof of Navier-Stokes has to be read by people who know the field, and for now the candidates for what falls next are being argued over on social media and at conferences and workshops [21].
What to watch
- Whether OpenAI names the second Millennium problem it says it has already made significant progress on.
- Whether anyone publishes a verification of the claimed Navier-Stokes solution, and who does the reading.
- Whether a candidate counterexample to the Birch and Swinnerton-Dyer criterion is submitted and survives checking.