AI model cracks number theory record by finding ultra-high-rank elliptic curves in days
Using Anthropic's language model Claude, mathematician Levent Alpöge and cryptographer Ava Howell discovered elliptic curves with ranks of at least 30 and 31, surpassing a record that had taken mathematicians over 18 years to advance from rank 28 to 29. These curves, defined by equations of the form y² = x³ + Ax + B, have rational points whose distribution becomes increasingly intricate as rank rises. The breakthrough demonstrates the potential of AI to accelerate discoveries in pure mathematics.
The rank of an elliptic curve reflects how many independent families of rational points exist on it. A rank-0 curve has finitely many such points, while rank-1 curves allow construction of all points from a single starting solution. Each additional rank adds another independent family, making the point distribution progressively more intricate.
The previous rank-29 record, set in August 2024, required mathematicians to construct cross-sections of higher-dimensional objects. Alpöge and Howell, both with deep expertise in elliptic curve research, obtained their rank-30 and rank-31 examples using a simple prompt directed at an internal, non-public Claude variant. The result follows OpenAI's recent claim of solving the Navier-Stokes problem.
This breakthrough could reshape how pure mathematics research is conducted, potentially accelerating fields where progress has historically been measured in decades. If AI models can reliably generate conjectures and constructions in number theory, researchers may redirect effort toward verifying and interpreting machine-produced results. However, the field's reliance on human intuition and proof verification means AI's role may remain complementary rather than replacement, with implications for funding priorities and collaboration between mathematicians and AI developers.