A neurosurgeon from Beijing, Shanmu Jin, has made headlines by utilizing AI to tackle a mathematical problem that has puzzled experts for over two decades. Jin, who has a background in geology and medicine but no formal training in mathematics, discovered the Crouzeix conjecture first proposed by French mathematician Michel Crouzeix in 2004 while delving into matrix analysis during his medical studies.
For 22 years, the conjecture remained unproven in general terms, although mathematicians had demonstrated it in specific instances and approached the necessary bounds.
The Crouzeix conjecture revolves around a square complex matrix A of size n × n. Its numerical range consists of all values represented by 〈 Ax, x 〉, where x is a complex unit vector. This range is a compact convex set in the complex plane, encompassing all eigenvalues of the matrix. The conjecture asserts that for any matrix and polynomial, the operator norm of the polynomial evaluated at the matrix should not exceed twice the maximum absolute value of that polynomial over the matrix's numerical range. Essentially, while the norm of the resultant matrix may surpass the polynomial's largest value, it can do so only by a factor of two, a constant that is universal irrespective of the matrix size or polynomial choice.
To approach this problem, Jin employed GPT-5.6 Sol, restricting its internet access and instructing it to deploy various strategies through multiple agents. He also directed these agents to critique each other's proofs until one method withstood rigorous examination. After allowing the AI to operate for roughly 16 hours without interference, it identified a crucial argument that formed the basis of the proof. Jin subsequently shared a preprint of the findings along with the prompt, intermediary versions, and formalization for the Lean proof-assistant system.
Though the paper has yet to undergo peer review, it has already garnered attention from mathematicians Alex Townsend, Anne Greenbaum, and even Michel Crouzeix, all of whom reviewed it thoroughly and deemed it valid. Remarkably, just eight days later, Dutch mathematicians Emiel Lorist and Felix Schwenninger also released an independent proof of the same conjecture, utilizing ChatGPT in their investigation.
This incident is not an anomaly. In May 2026, an OpenAI model successfully disproved the nearly 80-year-old Erdős conjecture regarding unit distances, a result later verified by independent mathematicians. Similarly, in August of that year, Anthropic employee Jarred Sumner asked Claude to attempt a proof of the Riemann hypothesis, one of the Millennium Prize Problems with a $1 million reward from the Clay Mathematics Institute. While the AI did not solve the conjecture, it significantly improved a related result, increasing the proportion of zeros of the zeta function that met the hypothesis's condition from 41.6% to 67.2%, a finding confirmed by Anthropic mathematicians and formalized in Lean.
The question remains: will AI eventually crack the Millennium Prize Problems? The consensus seems to lean towards optimism, with many believing that significant breakthroughs are on the horizon.
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