AI scores twin breakthroughs on decades-old math problems

In a span of weeks, artificial intelligence has notched two landmark results in pure mathematics: proving a conjecture that stumped specialists for more than 20 years and advancing one of the discipline's most celebrated unsolved problems.

A Doctor, a Chatbot, and a 22-Year-Old Conjecture

Jin Shanmu, a neurosurgery resident and postdoctoral researcher at the Peking Union Medical College Hospital in Beijing, proved Crouzeix's conjecture — a problem in numerical linear algebra posed by French mathematician Michel Crouzeix in 2004 — using OpenAI's ChatGPT. According to the South China Morning Post, Jin ran GPT-5.6-Sol autonomously for 16 hours on the ChatGPT Work platform, and the model produced a complete proof.scmp

Jin is not a trained mathematician. He studied geology as an undergraduate before attending medical school and was researching transcranial ultrasounds when he stumbled into the field of matrix analysis. The conjecture states that any function applied to a matrix produces a result no larger than twice the function's maximum value on that matrix's numerical range — an abstract but long-standing hurdle in the field.interestingengineering+1

Alex Townsend, a mathematician at Cornell University who had himself been prompting AI to tackle the problem, discovered on July 30 that ChatGPT informed him the conjecture had already been solved days earlier. Townsend reviewed Jin's manuscript and shared it with Anne Greenbaum of the University of Washington and Crouzeix himself, all of whom confirmed the proof was correct.interestingengineering

Claude Takes On the Riemann Hypothesis

Separately, Anthropic announced that an unreleased research version of its Claude model improved a bound tied to the 167-year-old Riemann hypothesis, raising the known lower bound for the fraction of zeros of the Riemann zeta function on the critical line from 41.6% to 67.2%. The work did not prove the hypothesis itself but represents a substantial advance on a related problem that had seen only incremental progress over decades.datacamp+1

The effort began when Jarred Sumner, an Anthropic employee, prompted Claude to attempt the Riemann hypothesis. After trying roughly 650 ideas over several days without success, Sumner encouraged the model with messages like "keep going" and "believe in yourself." Claude then spent a day and a half coordinating with about 60 copies of itself, consuming 31 million output tokens, before landing on the result. According to The Wall Street Journal News Corp ↗ 0.73%, a Stanford number theorist called it "the most remarkable result that AI has generated in mathematics to date".wsj+1

A Pattern Emerging

These results follow OpenAI's May announcement that an internal model disproved the Erdős unit-distance conjecture, a problem dating to 1945. In late July, OpenAI also published a list of ten mathematical and computer science advances made by its models. The rapid succession of breakthroughs lends weight to mathematician Terence Tao's prediction that 2026-level AI would become "a trustworthy co-author in mathematical research".phys+3