The most complicated result to date in the field of number theory looks deceptively simple: it’s essentially a curve and a line. More technically speaking, it’s an elliptic curve with a rank of at least 31—meaning it’s a special mathematical curve with at least 31 independent starting points for meaningful patterns. The higher the rank, the more complex the pattern, which determines how certain points are distributed along that curve.
For decades, mathematicians have been chasing elliptic curves with ever more complicated patterns and have made very slow progress. It took more than 18 years to get from rank 28 to rank 29, for example. But now mathematician Levent Alpöge, who works at the artificial intelligence company Anthropic, and cryptographer Ava Howell have used Anthropic’s large language model Claude to present both an elliptic curve with a rank of at least 30 and one with a rank of at least 31 within just a few days.
The underlying elliptic curves of the equations seem quite simple—they have the form y2 = x3 + Ax + B. And yet they have occupied mathematicians for millennia. Of particular interest is the distribution of “rational points” on the curves—that is, those valuesof x and y that can be expressed as a fraction and satisfy the equation.
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Elliptic curves of rank 0, for example, have only a finite number of rational points. Rank 1 curves have infinitely many, but the points follow a simple pattern. If you know one of them, you can work your way to any other with simple tricks that involve reflecting, projecting and connecting the points with lines. Thus, from a single rational solution, all other rational points can be constructed.
The points of elliptic curves have a remarkable structure: they can be transformed into one another. These transformations form a group.
Amanda Montañez
For an elliptic curve of rank 2, there are two independent families of rational points: if you know one rational point from a family, you can construct all the other points from that family—but not the points from the other family. So you need two points—one from each family—to get to all the other points on the curve. Similarly, elliptic curves of degree 3 have three independent families, and so on.
A key open question in number theory is whether elliptic curves can have infinitely high degrees—or whether there is a limit. Experts are searching for examples of elliptic curves with ever higher degrees. Most recently, after an 18-year hiatus, an elliptic curve with a degree of at least 29 was discovered in August 2024. To construct this curve, experts formed a cross section of objects in higher dimensions, the boundary of which corresponds to the elliptic curve.
And now, just two years later, an AI language model has provided two examples of elliptic curves of an even higher rank. Alpöge and Howell—who both have extensive background knowledge in the field of elliptic curve research—obtained this result using a relatively simple prompt. The language model they used is an internal variant of Claude that is not publicly available.
The news comes on the heels of the stunning announcement last week from OpenAI that its own model had solved the Navier-Stokes problem, one of the seven Millennium Prize Problems.
This article originally appeared in Spektrum der Wissenschaft and was reproduced with permission. It was translated from the original German version with the assistance of artificial intelligence and reviewed by our editors.
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