Anthropic Claude Shatters 18-Year Pure Mathematics Record: Discovers Most Complex Elliptic Curve with Rank 31

In pure mathematics, few quests have proven as stubbornly resistant to brute computation as the search for high-rank elliptic curves. For decades, the world's leading number theorists have competed to construct explicit algebraic curves possessing the highest possible number of independent rational solutions.

Now, in a historic milestone for automated scientific discovery, Anthropic mathematician Levent Alpöge and cryptographer Ava Howell have shattered that record. Utilizing an internal frontier variant of Claude, the researchers discovered elliptic curves with algebraic ranks of at least 30 and 31 over the rational numbers—surpassing the previous world record of rank 29 established in August 2024.

What makes the achievement truly extraordinary is not merely the mathematical object itself, but the velocity of its discovery. Moving from rank 28 to rank 29 took the human mathematical community more than 18 years of specialized algebraic geometry and massive distributed computer searches. Claude's new rank-30 and rank-31 curves emerged within days from relatively simple, structured exploratory prompting.

Elliptic curves, typically expressed in the Weierstrass canonical form y² = x³ + Ax + B (where A and B are integers), are foundational structures in modern mathematics and modern cryptography. They formed the core mechanism behind Andrew Wiles's celebrated proof of Fermat's Last Theorem and provide the mathematical bedrock for elliptic-curve cryptography securing global internet communications and blockchain networks.

Under the Mordell-Weil theorem proven in the 1920s, the set of points on an elliptic curve whose coordinates are rational numbers forms a finitely generated abelian group. This group decomposes into a finite torsion subgroup (points of finite order) and an infinite free part generated by a finite number of independent rational points. The number of these independent generators is known as the curve's algebraic rank.

Geometrically, a curve with rank zero has only finitely many rational points. A curve with rank one has an infinite line of points generated by adding a single rational point to itself via the curve's chord-and-tangent group law. A curve of rank 31 possesses 31 completely independent, non-repeating rational directions across its geometric surface—creating an astonishingly intricate web of fractional solutions.

Finding curves with large rank is famously difficult because high-rank curves are exceedingly rare. In 2006, legendary mathematician Noam Elkies of Harvard University set the long-standing record by discovering an elliptic curve of rank at least 28. For nearly two decades, exhaustive computing projects and algorithmic sieves failed to push the frontier further.

In August 2024, Elkies and Zev Klagsbrun finally announced the discovery of a curve with rank at least 29, requiring immense mathematical ingenuity and complex specialized search software.

Alpöge, a former Junior Fellow in the Harvard Society of Fellows who recently joined Anthropic, partnered with Howell to test whether frontier large language models could reason about elliptic curve constructions. Rather than relying on rigid brute-force search scripts, the researchers tasked an internal Claude variant with exploring specialized polynomial families and constructing parameter spaces likely to force rational points.

The model proposed algebraic constructions that yielded explicit curves with rank at least 30, and subsequently pushed the frontier to rank at least 31. The discovery was initially submitted to the global elliptic curve rank records database under the pseudonymous handle "ranksunbounded," before the authors added a confirming note that the discovery was driven by Claude working alongside Alpöge and Howell.

The exact rank of 31 for the new curve has been rigorously verified subject to the Birch and Swinnerton-Dyer (BSD) conjecture—one of the seven Millennium Prize Problems—and the Generalized Riemann Hypothesis (GRH). Conditional rank calculations using modern computer algebra systems (such as PARI/GP and SageMath) confirm the presence of 31 independent rational generators.

The result injects fresh momentum into one of the deepest open debates in modern number theory: the Unbounded Rank Conjecture. For generations, mathematicians hypothesized that elliptic curves over Q could possess arbitrarily high ranks. However, heuristic models proposed over the past decade by number theorists (including Watkins and Poonen) suggested that rank might actually be bounded by an upper threshold, with higher ranks representing vanishingly rare statistical anomalies.

While Claude's discovery of a rank-31 curve does not mathematically resolve whether ranks can grow infinitely, it demonstrates that the practical computational ceiling can be pushed significantly higher through machine-assisted mathematical intuition.

For the artificial intelligence industry, the discovery represents a profound qualitative inflection point. Until now, LLM mathematical capabilities were largely measured on static competition benchmarks such as the International Mathematical Olympiad (IMO), Putnam exams, or synthetic theorem-proving environments in Lean.

In this instance, Claude was not solving a problem with a known textbook answer. It was actively exploring the unknown, formulating non-trivial algebraic conjectures, and isolating mathematical objects that world-class human specialists had failed to find over two decades.

As frontier AI laboratories continue developing reasoning models equipped with automated verification sandboxes, the intersection of deep linguistic reasoning and formal mathematics is poised to accelerate scientific breakthroughs from pure number theory to quantum chemistry and cryptographic design.

• Scientific American - Anthropic’s AI steals mathematicians’ record for most complicated curve
• Superpower Daily - Anthropic Researchers Use Internal Claude to Set a Higher Elliptic-Curve Record
• ForkLog - Claude Helps Set Record for Elliptic Curves
• Epoch AI FrontierMath - Elliptic Curves over Q of Large Rank Open Benchmark