Chinese doctor cracks decades-old math problem using ChatGPT

Jin Shanmu, a postdoctoral researcher and resident physician at Peking Union Medical College Hospital in Beijing, proved Crouzeix's conjecture, a long-standing problem in numerical linear algebra, with the help of GPT-5.6 Sol, an AI model available through ChatGPT Work, according to the South China Morning Post.

Proposed by French mathematician Michel Crouzeix in 2004, the conjecture concerns how functions behave when applied to matrices. In simplified terms, it proposes a mathematical limit on how large the result can become based on a matrix's numerical range.

The highly abstract problem has long attracted researchers in matrix analysis. Jin encountered the field while conducting research involving transcranial ultrasound.

Alex Townsend, a mathematics professor at Cornell University in the U.S., had spent more than a year using GPT models to search for a solution to Crouzeix's conjecture. On July 30, one of the models directed him to Jin's newly published proof.

After examining the work, Townsend and Anne Greenbaum, a mathematics professor at the University of Washington, U.S., were stunned by the result. Crouzeix also confirmed that Jin's proof was correct.

A person uses ChatGPT on a smartphone. Illustration from Pexels

The breakthrough highlighted "how strikingly capable" GPT-5.6 had become in recent weeks, particularly given Jin's lack of specialized mathematical training, Townsend and Greenbaum wrote.

Jin studied geology as an undergraduate before switching to medicine. In an email to the mathematicians, he said his formal mathematics education was limited to basic undergraduate courses for science students and that he was otherwise self-taught.

A key part of Jin's approach was how he instructed the AI to tackle the problem, according to online news site 36Kr Europe.

Rather than simply asking ChatGPT for an answer, Jin prompted it to operate more like a team of mathematicians exploring and testing competing ideas. He instructed the model to work without Internet access, pursue multiple approaches rather than settling quickly on one, and repeatedly challenge potential solutions by searching for errors and counterexamples.

The AI was also told to continue working until it produced a complete proof capable of withstanding rigorous scrutiny. After setting up the instructions, he started the process and returned to his clinical work, leaving the AI to operate largely without his intervention.

GPT-5.6 Sol worked for about 16 hours, cycling through tens of thousands of hypotheses, rejections and reformulations before eventually completing the proof.

Jin's achievement comes as frontier AI systems play an increasingly prominent role in advanced mathematical research. In May, U.S. firm OpenAI said an internal general-purpose reasoning model had autonomously tackled an 80-year-old planar unit distance problem posed by Hungarian mathematician Paul Erdos in 1946.

Earlier this month, OpenAI published a list of 10 additional mathematical problems that an internal version of its next major model, Astra, had either resolved or made significant progress on.

U.S. AI company Anthropic also said Monday that an unreleased research version of its Claude model had attempted to solve the Riemann hypothesis, one of mathematics' most famous unsolved problems. Although Claude failed to prove the hypothesis, the company said the model had "unexpectedly made strides" on a related problem.