Anthropic's Claude Helps Find an Elliptic Curve of Rank at Least 31

A long-running hunt in number theory has a new reported high-water mark. Anthropic mathematician Levent Alpöge and cryptographer Ava Howell used an internal Claude variant to find elliptic curves with ranks of at least 30 and 31, exceeding the previously reported rank-29 example.

The result lands after an unusually slow stretch in the field. The move from rank 28 to rank 29 took more than 18 years, according to Scientific American. That rank-29 curve was discovered in August 2024; the newly reported rank-31 curve goes two ranks higher.

A simple-looking equation, a hard search

The curves in question take the form y² = x³ + Ax + B. What makes them difficult is the arrangement of their rational points: solutions whose coordinates can be expressed as fractions. Rank counts independent families of those points. More families mean a more complex pattern for mathematicians to map.

Alpöge and Howell have extensive background in elliptic-curve research. Still, the reported pace is striking: Scientific American says the two examples emerged within a few days from a relatively simple prompt. The account does not describe a public Claude release or a broadly available mathematical tool.

The next question is larger than the record

The Claude variant used in the work is not publicly available. That limits who can directly use the same system while the result enters a field focused on a deeper unresolved question: whether elliptic curves can have arbitrarily high ranks, or whether a limit exists. The new examples do not answer that question, but they supply higher points in the search.