AI Solves 85-Year-Old Jacobian Conjecture

An artificial intelligence model has successfully resolved the Jacobian conjecture, a complex mathematical problem that has eluded mathematicians since 1939. The breakthrough was announced on a Sunday afternoon and verified by the following morning, sparking widespread discussion among mathematicians. Kevin Buzzard of Imperial College London described it as a "big day" and a "great time to be alive."

This achievement follows a rapid acceleration of AI's impact on mathematics since mid-2025, when AI models first solved multiple problems at the International Mathematical Olympiad. More recently, OpenAI's AI disproved an 80-year-old Erdős conjecture in combinatorial geometry in May. These advancements have prompted concerns within the mathematical community, leading to the publication of the Leiden Declaration in June by 16 researchers from 15 universities. This declaration calls for guardrails on transparency, attribution, and peer review as AI continues to reshape mathematical knowledge.

The Jacobian conjecture, named after German mathematician Ott-Heinrich Keller, concerns the properties of "maps" and the conditions under which inputs can be determined from outputs. Its foundation dates back to the work of Carl Gustav Jacob Jacobi a century prior. The conjecture has been a persistent challenge for modern mathematicians due to the difficulty in proving its core tenets.

The reaction from mathematicians is a blend of awe and apprehension, as AI models demonstrate an increasing capacity to tackle problems previously considered intractable for human intellect. This latest resolution represents a significant milestone in AI's growing capabilities in abstract reasoning and problem-solving, pushing the boundaries of what was once thought possible.