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Anthropic Researchers Use Internal Claude to Set a Higher Elliptic-Curve Record

The reported rank-31 result advances a search that had needed more than 18 years to move from rank 28 to rank 29, but the Claude variant behind it is not public.

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Anthropic Researchers Use Internal Claude to Set a Higher Elliptic-Curve Record
Anthropic Researchers Use Internal Claude to Set a Higher Elliptic-Curve Record

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Anthropic researchers have reported a new high-water mark in the search for complicated elliptic curves: examples with ranks of at least thirty and thirty-one, found with help from an internal version of Claude. The result goes beyond the rank-twenty-nine curve reported in August 2024—and it arrived far faster than the field’s recent pace. Moving from rank twenty-eight to rank twenty-nine had taken more than eighteen years, according to Scientific American. The new examples reportedly emerged within a few days from a relatively simple prompt. The researchers were Anthropic mathematician Levent Alpöge and cryptographer Ava Howell, both of whom have extensive experience with elliptic-curve research. The underlying objects can look deceptively simple. They follow an equation read as: y squared equals x cubed, plus A x, plus B. The challenge is in their rational points—solutions whose coordinates can be written as fractions. Rank measures the number of independent families of those points, so higher rank indicates a more intricate structure for mathematicians to study. But this is a record in a search, not a solution to the field’s biggest question. The examples do not show whether elliptic curves can reach arbitrarily high ranks or whether some upper limit exists. And because the Claude variant is not publicly available, other researchers cannot yet reproduce the search with the same system. The key thing to watch is whether this inaccessible result leads to further records—or remains a striking but difficult-to-verify data point.

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3 key points

An internal Claude variant helped Anthropic researchers produce elliptic curves with ranks of at least 30 and 31, extending the reported record beyond the rank-29 example found in August 2024. Scientific American reports the new examples emerged within days from a relatively simple prompt, despite an 18-year gap between earlier records. The result demonstrates a potentially useful research capability, but its...

  1. 01

    The new curves reach ranks 30 and 31, surpassing the previously reported rank-29 record.

  2. 02

    The jump from rank 28 to rank 29 took more than 18 years.

  3. 03

    Anthropic mathematician Levent Alpöge and cryptographer Ava Howell reportedly found the examples within days.

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.

Sources

  1. scientificamerican.comAI steals mathematicians’ record for most complicated curve

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Anthropic Researchers Use Internal Claude to Set a Higher Elliptic-Curve Record | Superpower Daily