A model from Anthropic made progress on one of the major problems in mathematics

Anthropic reported that an unpublished model has made progress on the Riemann hypothesis, a unsolved problem for over 150 years.

The system tested 650 ideas over a day and a half, coordinated 60 subagents, and used 31 million tokens to search for a solution.

What the Anthropic model achieved with the Riemann hypothesis

The Riemann hypothesis is one of the major open problems in mathematics and is linked to the distribution of prime numbers.

Anthropic reported that a model not yet publicly presented managed to significantly increase the lower bound of solutions for which the hypothesis holds.

The result does not constitute a general proof of the hypothesis. Currently, there is a reward of US$1 million for a valid and complete proof.

How artificial intelligence worked to advance the problem

A member of Anthropic without significant mathematical training asked the model to try to solve the problem and then let it organize the work for a day and a half.

During that period, the system tested 650 different ideas and coordinated the work of 60 subagents specialized in various tasks.

• 2 subagents developed the main mathematical ideas.

• 13 subagents contributed ideas for those agents.

• 30 subagents attempted to develop new ideas without success.

The proof was formalized with Lean

The work was also formalized using Lean, an open-source theorem proving assistant.

This instance allowed for the formal verification of the result presented by the Anthropic model, as explained by the company.

AI adds new results in mathematics

The progress on the Riemann hypothesis adds to other mathematical results recently obtained through large-scale language models.

During this year, various artificial intelligence systems solved a series of Erdős problems. Additionally, OpenAI presented ten significant mathematical results obtained by its internal model Astra.

Anthropic also contributed to these advancements with a separate work that managed to refute the Jacobian conjecture, according to the information provided.

The debate among mathematicians over AI's advancement

These results have also sparked a debate within the mathematical community. A public statement signed in June raised concerns about the impact of AI on mathematics.

Among the points raised is the importance of mathematical proofs being attributable to specific authors, responsible for both their discoveries and their correctness.

However, the position is not unanimous. Fields Medal winner Timothy Gowers questioned this perspective and suggested that the influence of AI could transform mathematics in a more complex and also positive way.

In total, the process required 31 million output tokens. The result was confirmed by two internal mathematicians at Anthropic.