AI Agents Failed to Prove Fermat's Last Theorem. Then They Got a Shared To-Do List

On September 4, Anthropic announced a groundbreaking achievement: the first complete, computer-checked proof of Fermat's Last Theorem, authored by a team of autonomous Claude agents over 11 days. To truly grasp the significance, three crucial facts stand out.

Firstly, the proof itself was monumental. It consisted of 13 million lines of Lean, nearly 30,000 intermediate theorems, and generated about 6 billion output tokens. This is a stark contrast to Mathlib, a community proof library that serves as the foundation for such proofs, which is relatively concise and well-reviewed.

Secondly, the agents achieved this feat by employing a shared directed acyclic graph (DAG) as their memory. This coordination layer proved essential, as the initial attempts failed due to the agents losing track of the project's state and no longer collaborating effectively. This failure was not a result of the model's limitations; rather, it was the absence of a shared state management system that led to the downfall.

The fix came in the form of Prove2Me, an open platform designed by Tianyi Peng and his team at Columbia University. This platform addressed three key issues. Firstly, it maintained a DAG of theorem statements, allowing agents to consult it when deciding what to prove next. This helped mitigate the degradation of context windows, which are limited in size and can cause agents to lose track of information.

Secondly, it separated theorem statements from their proofs into different files, with the links maintained independently. This improved Lean compilation and reduced resource consumption. Lastly, it kept a natural-language description of every node, enabling search and reuse. This feature allowed later agents to find existing results instead of re-deriving them, resulting in a simpler proof path.

The success of Prove2Me's approach is evident in the numbers. While the failed attempts accounted for about 7% of the non-boilerplate lines in the final proof, the corrected approach led to the successful completion of the task. This story underscores a fundamental lesson applicable to our daily workflows: shared state management is crucial.

When agents hold shared state in their context, they become time bombs, prone to making the same mistakes repeatedly. By externalizing coordination into persisted checklists and structured state, we can prevent such errors and achieve more reliable and consistent results.