Claude and mathematicians Levent Alpöge and Ava Howell found an elliptic curve with 31 independent points: the ICARM database verified that its rank is at least 31

Claude and mathematicians Levent Alpöge and Ava Howell found an elliptic curve with 31 independent points: the ICARM database verified that its rank is at least 31

On August 23, entry №302 appeared in the ICARM mathematical database, containing a new equation and 31 independent rational points. This establishes a new unconditional lower bound of 31 for the curve's rank.

Here, an elliptic curve is an equation whose rational solutions can be combined using a particular addition rule. The rank measures the number of independent directions among these solutions. A point does not add a new direction if it can be obtained by adding the others. The 31 points in entry №302 therefore provide the evidence to be checked, rather than merely a number for the headline.

ICARM is a database of such curves. It checks that every submitted point lies on the stated curve, then proves that the points are independent using an exact mathematical procedure called 2-descent. The calculation involves no numerical rounding. This procedure connects the equation and list of points in the entry to the conclusion that the curve's rank is at least 31.

The FrontierMath problem from Epoch AI required an equation and 30 independent points. On August 20, entry №273, which described a different curve, raised the known lower bound from 29 to 30. Epoch AI attributed that result to Anthropic's internal Claude model, Levent Alpöge, and Ava Howell. Three days later, the same team received attribution for №302.

The FrontierMath threshold was 30. Entry №302 contains a different equation and 31 points, so it raises the lower bound by one more step.