AI Models Discover Counterexamples to Long-Standing Mathematical Conjectures

ME AI News: According to monitoring by Beating, Anthropic mathematician Levent Alpöge used Claude Fable to find a three-dimensional counterexample to the Jacobian Conjecture. This problem, unresolved since 1939, is listed among the important mathematical challenges of the 21st century. The polynomial mapping provided by Fable has a Jacobian determinant constantly equal to −2, yet maps three distinct inputs to the same output, proving it cannot be invertible—thus falsifying the three-dimensional and higher-dimensional versions, while the two-dimensional case remains unsolved. Subsequently, OpenAI researcher Aaron Lou asked the internal Codex model to independently attempt the same task. Without internet access, the model also discovered an essentially identical counterexample and derived the reasoning process. In the future, mathematicians may simultaneously submit numerous open problems to different models. AI can iteratively test hypotheses to find counterexamples, vulnerabilities, and new insights, then verify results using symbolic computation or formal proof tools. Many directions previously too time-consuming to exhaustively explore may now be rapidly surveyed, potentially accelerating scientific discovery. However, problems will not automatically become simpler. Models will also generate batches of seemingly rigorous but incorrect proofs, shifting the academic challenge from “unable to find answers” to “unable to verify answers in time.” The greater risk lies in genuine breakthroughs: RSA security relies on the difficulty of factoring large integers; Diffie-Hellman and elliptic curve cryptography depend on the hardness of solving discrete logarithms. If AI discovers a sufficiently fast new algorithm, parts of public-key systems used in website encryption, digital signatures, banking transactions, and blockchains could become invalid, forcing a global emergency overhaul of cryptographic standards. (Source: BlockBeats)