For over a century and a half, the Riemann hypothesis has remained one of mathematics' most celebrated open questions—an enduring puzzle centered on how prime numbers are distributed. A complete, general proof of the conjecture carries a $1 million prize, and so far, no one has claimed it. Today's AI models can't crack it either, but they're capable of making far more headway than many would assume, a development that's poised to reignite debates about whether contemporary artificial intelligence can genuinely generate new scientific and mathematical insights.

On Monday, Anthropic revealed that one of its unreleased models achieved meaningful progress on the Riemann hypothesis, notably pushing upward the lower bound for which the conjecture is known to hold. What makes the achievement stand out is the method behind it. An Anthropic employee, lacking any deep mathematical background, simply asked the model to "take a real stab" at proving the hypothesis, then stepped back while the system managed the task over the next 36 hours. During that period, the model explored 650 distinct approaches, coordinating across 60 sub-agents and allocating 31 million computational steps in total. A footnote in the accompanying paper breaks down the division of labor: "Out of the 60 subagents, two were responsible for developing the key mathematical ideas, 13 contributed ideas to these agents, 30 attempted (but were unable) to develop new ideas, 13 served as validators to check the correctness of the arguments, and the final two helped to write the initial paper."

Two of Anthropic's in-house mathematicians verified the findings, and the results were formally encoded using Lean, an open-source proof assistant. This marks the latest in a growing series of mathematical advances driven by large language models (LLMs). Throughout this year, AI systems have resolved several Erdős problems, and the arrival of more capable models has produced increasingly notable outcomes. OpenAI recently published ten major results proven by its internal "Astra" model, while a separate Anthropic effort refuted the long-standing Jacobian conjecture. These accumulating successes have stirred both enthusiasm and unease within the mathematical community. In a public statement signed in June, a group of prominent mathematicians voiced concerns that AI might erode key values of the discipline—particularly the expectation that genuine proofs be "attributable to specific authors who take credit for their discovery and assume responsibility for their correctness."

Yet the field remains divided on how to embrace these new research methods. Responding to the declaration in a blog post, Fields Medalist Timothy Gowers suggested that AI's influence could reshape mathematics in ways that are more nuanced and potentially beneficial. "If we arrive at a world where mathematical theorems are no longer associated with mathematicians, maybe that won't be any more problematic than the fact that stars aren't named after astronomers and most aren't named at all," Gowers wrote.