Anthropic Model Makes Progress on the Riemann Hypothesis

The Riemann hypothesis has remained one of the great unsolved problems in mathematics for more than 150 years now: it concerns the distribution of the prime numbers, and a $1 million prize has been set aside for a general proof — a prize that remains unclaimed to this day. Today's AI models haven't solved it either — but, as it turns out, they can push further than one might have expected.

On Monday, Anthropic announced that one of its as-yet-unreleased models had substantially extended the lower bound on the values for which the hypothesis is confirmed. The result was verified by two of the company's in-house mathematicians, and the argument itself was formalized using the open-source proof assistant Lean.

How it was done

The process itself is no less intriguing. An Anthropic employee with no serious mathematical background asked the model to “seriously try” to prove the hypothesis, then left it to coordinate the work over the course of a day and a half. In that time, the model worked through 650 different ideas, distributing the task across 60 subagents and consuming a total of 31 million output tokens.

A footnote to the article offers a telling “breakdown of roles”: of the 60 subagents, two were responsible for the key mathematical ideas, 13 fed them hints, another 30 tried (unsuccessfully) to come up with something new, 13 acted as checkers, and the final two wrote the initial draft of the paper.

Not the first and not the only case

This is merely the latest result in a string of mathematical achievements by large language models. Over the past year, AI has solved a number of Erdős problems, OpenAI presented a dozen results proved by its internal model “Astra,” and another team at Anthropic disproved the long-standing Jacobian conjecture.

The accumulating successes have stirred both excitement and anxiety in the mathematical community. In a statement published in June, a group of prominent mathematicians warned that AI could undermine important values of the discipline — first and foremost the idea that a proof should belong to specific authors who put their name to a discovery and answer for its correctness.

There is as yet no consensus on how to regard the new methods. In a response note, Fields Medalist Timothy Gowers suggested that AI's influence might change mathematics in a more complicated but positive direction. “If we end up in a world where theorems are no longer tied to the names of mathematicians, perhaps that will be no more problematic than the fact that stars aren't named after astronomers, and most aren't named at all,” he wrote.

Source: TechCrunch