AI Begins to Tackle Mathematical Challenges, Claude and Codex Successively Find Counterexamples to Unsolved Conjectures

According to Botify Beating monitoring, Anthropic mathematician Levent Alpöge used Claude Fable to find a three-dimensional counterexample to the Jacobian Conjecture. This problem has been unresolved since 1939 and is still listed as one of the important mathematical challenges of the 21st century.

The polynomial mapping provided by Fable, with a Jacobian determinant always equal to -2, maps three distinct inputs to the same output. Therefore, it is inherently non-invertible. While the three-dimensional and higher-dimensional versions have been disproven, the two-dimensional version remains unsolved.

Subsequently, OpenAI researcher Aaron Lou had the internal Codex independently attempt the problem. The model, even in an offline setting, found a fundamentally similar counterexample and produced the derivation process.

In the future, mathematicians may entrust a large number of open problems to different models simultaneously. AI can iterate through trial and error, searching for counterexamples, loopholes, and new insights, and then use symbolic computation or formal proof tools to validate. Many directions that were previously too time-consuming for exhaustive exploration may now be rapidly surveyed, potentially accelerating scientific discoveries significantly.

However, this does not automatically make problems easier. Models may also mass-produce seemingly rigorous incorrect proofs, shifting the academic community's focus from "unable to find an answer" to "having no time to validate answers."

A greater risk comes from a real breakthrough. The security of RSA relies on the difficulty of factoring large integers, while Diffie-Hellman and elliptic curve cryptography rely on the computational infeasibility of discrete logarithms. Once AI discovers a new algorithm fast enough, parts of the public key systems used in website encryption, digital signatures, bank transactions, and blockchain could all become vulnerable, requiring a global emergency transition to new cryptographic standards.