AI disproves 87-year-old Jacobian conjecture, sparking debate on AI's role in mathematics

AI Disproves Mathematical Conjecture That Stood for Nearly a Century

While millions watched the World Cup final on Sunday evening, mathematician Levent Alpöge casually announced on X that he had found a counterexample to the Jacobian conjecture, an 87-year-old problem that has challenged mathematicians since 19392

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. The breakthrough came with help from Anthropic's Fable 5, a large language model released just weeks earlier3

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. The counterexample was remarkably concise at just 216 characters, short enough to fit in a single social media post4

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. Alpöge, a Harvard fellow with an affiliation to Anthropic, thanked his "close friend fable" for working during the match, revealing AI in mathematics had achieved what decades of human effort could not2

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Understanding the Long-Standing Problem in Algebraic Geometry

The Jacobian conjecture, formally set out by German mathematician Ott-Heinrich Keller in 1939, involves polynomial functions that map points in space2

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. It states that when a function's Jacobian determinant is a non-zero constant, there should always exist another polynomial function that reverses the original one, returning all points to their starting positions3

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. The problem was so compelling that Fields Medallist Stephen Smale included it in his influential 1998 list of Mathematical Problems for the Next Century3

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. Throughout its history, the conjecture attracted many claimed proofs, including attempts by famed 20th-century mathematicians Beniamino Segre and Wolfgang Gröbner, but subtle errors invalidated each argument3

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AI's Role in Pure Mathematics Expands Beyond Graph Theory

Alpöge's discovery represents the biggest mathematical conjecture where AI has played a significant role in proving or disproving so far, according to Abhishek Saha at Queen Mary University of London2

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. The finding joins a growing series of AI-driven mathematical breakthroughs. Just days earlier, ChatGPT cracked the 30-year-old Dinitz-Garg-Goemans conjecture in graph theory using extremely basic prompts totaling fewer than 60 words over five and a half hours1

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. In May, an OpenAI model disproved the Erdős unit distance conjecture, a central problem in discrete geometry5

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. These rapid advances have created a website cataloguing AI findings and listing them by the model used1

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Debate Emerges Over What Constitutes Mathematical Progress

Not all mathematicians view disproving the Jacobian conjecture as transformative. Andrew Blumberg at Columbia University, who works on testing AI capabilities in mathematical research, was pointedly unmoved by the result5

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. "This did not cause me to update my priors about what AI can and can't do," he told Mashable, noting this is exactly what he expects AI to excel at5

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. He emphasized a crucial distinction: finding a counterexample differs fundamentally from building proofs that reveal why something is true. "There are a lot of polynomials, and it is hard for people to check them all, but it is not hard for machines," Blumberg explained5

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. Chris Bowman-Scargill at the University of York agrees that while AI has proven it can find counterexamples, building whole new branches of mathematics still requires human creativity2

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Short-Term Impact on Mathematical Research Practices

The brevity and simplicity of Alpöge's counterexample made it straightforward for other mathematicians to verify, though the process of finding it required genuine insight beyond simple prompting4

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. Saha notes that while the result is easy to check, the method remains unclear: "I don't know how he did it, what exactly was the prompt to give Fable, because if one were to search everything, it wouldn't quite work, so obviously there was some insight also which is not currently published"2

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. The finding shows the conjecture is false for every dimension larger than two, with the original two-dimensional version remaining open3

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. OpenAI researcher Aaron Lou revealed an internal version of the company's Codex model found essentially the same counterexample independently4

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Future Implications for Researchers and AI Capabilities

Alexander Yong at the University of Illinois Urbana-Champaign suggests AI's growing role will help researchers discard dead ends and pursue promising directions instead1

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. "Those conjectures that survive AI scrutiny will be the genuine goals for human innovation," Yong noted1

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. Current AI models seem particularly well-suited to problems like finding counterexamples, though there are limits to what is currently possible, and the problems solved by AI so far are of limited complexity1

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. Ivan Fesenko at Westlake University in China predicts increasingly capable AI models will solve ever more complex problems, fundamentally changing mathematics: "Right now, AI can already produce master's degrees in mathematics. In one year, they will produce PhD degrees in mathematics"2

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. Alpöge hints the story may not end in destruction, suggesting his counterexample might reveal a positive result hiding inside, with a full write-up to follow4

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