Anthropic's Claude AI Finds Elliptic Curves of Rank 31

Anthropic's Claude language model has achieved a breakthrough in pure mathematics, discovering elliptic curves with ranks of at least 30 and 31—a feat that represents years of painstaking human effort compressed into days.

Levent Alpöge, a mathematician at Anthropic, and cryptographer Ava Howell used an internal variant of Claude to identify these highly complex mathematical structures. The discovery advances a field where progress had been glacially slow: it took more than 18 years for mathematicians to move from rank 28 to rank 29, with rank 29 only discovered in August 2024.

Understanding elliptic curve rank

Elliptic curves appear deceptively simple, taking the form y² = x³ + Ax + B. What makes them fascinating to mathematicians is the distribution of "rational points"—values of x and y that can be expressed as fractions and satisfy the equation.

The rank of an elliptic curve indicates how many independent families of rational points exist on it. A rank 0 curve has only finitely many rational points. A rank 1 curve has infinitely many points, but they follow a predictable pattern—knowing one point allows you to construct all others through geometric operations.

Higher ranks mean greater complexity. A rank 2 curve has two independent families of points; you need one point from each family to reach all other points on the curve. This pattern continues: rank 3 curves have three independent families, and so on. The newly discovered rank 31 curve has at least 31 such independent starting points.

How Claude achieved the breakthrough

Alpöge and Howell, both with extensive backgrounds in elliptic curve research, obtained the result using what they describe as a relatively simple prompt to Claude. The specific variant they used is not publicly available.

The previous record—rank 29—was achieved through sophisticated techniques involving cross-sections of higher-dimensional objects. Claude's ability to identify rank 30 and 31 curves in days suggests AI models may be capable of pattern recognition or search strategies that complement traditional mathematical approaches.

Why it matters

This achievement demonstrates AI's potential to accelerate progress on fundamental mathematical problems that have resisted human effort for decades. A central open question in number theory is whether elliptic curves can have infinitely high ranks or whether a maximum exists. Finding curves of ever-higher rank provides crucial data points for this question.

The result also arrives amid growing evidence of AI capabilities in advanced mathematics. Scientific American notes this follows OpenAI's recent announcement that its model solved the Navier-Stokes problem, one of seven Millennium Prize Problems.

For the mathematics community, the discovery raises questions about how AI tools might reshape research workflows—not replacing human mathematicians, but potentially serving as powerful collaborators in exploring mathematical landscapes that are computationally vast but conceptually structured.

The findings were first reported by Scientific American, based on original reporting from Spektrum der Wissenschaft.