A mathematician at Anthropic has used the company's Claude AI to disprove the Jacobian conjecture, a problem that had stumped researchers for 87 years. The result, confirmed this week, has immediate implications for cryptographic systems that rely on the hardness of polynomial equations.
The 87-year-old problem
The Jacobian conjecture, first posed in 1939, asks whether a polynomial map with a non-zero constant Jacobian determinant must have a polynomial inverse. For decades, mathematicians believed it was true, but no one could prove it. The conjecture sat in the same category as other famous unsolved problems — widely assumed but never settled.
Anthropic's mathematician, whose name has not been released, turned to Claude AI to explore potential counterexamples. The AI helped construct a specific polynomial map that violates the conjecture's conditions, effectively disproving it. The work was verified by independent mathematicians this month.
How Claude changed the game
This isn't the first time AI has helped solve a math problem, but it's one of the most significant. The Jacobian conjecture touches algebraic geometry and polynomial algebra — fields where human intuition has long been the only tool. Claude didn't just brute-force search; it generated candidate maps and then helped reason through the algebraic structure.
The mathematician told colleagues that without Claude, the counterexample would have taken years to find, if ever. The AI's ability to explore vast combinatorial spaces and suggest non-obvious patterns was key.
Crypto systems in the crosshairs
The disproof raises questions about AI's role in cryptographic systems. Many post-quantum cryptography schemes, especially those based on multivariate polynomials, assume that certain polynomial problems are hard. If an AI can disprove a foundational conjecture like Jacobian, it could also find weaknesses in those assumptions.
Cryptographers are now re-examining the security margins of algorithms that rely on polynomial inversion or related problems. The timing isn't great — the National Institute of Standards and Technology is still finalizing its post-quantum standards, and any doubt about the underlying math could delay adoption.
What the math community is saying
Reactions have been mixed. Some mathematicians see the disproof as a triumph of human-AI collaboration. Others worry that AI could destabilize fields built on unproven conjectures. The Jacobian conjecture was never a direct cryptographic primitive, but its disproof shows that AI can upend long-held beliefs.
Anthropic has not commented on whether it will release the full proof or the AI-generated counterexample. The cryptographic community is watching closely — and waiting for the next shoe to drop.
