Anthropic PBC has utilized its AI model, Claude, to generate a computer-verifiable version of a historically complex mathematical proof. The company released details of this significant project in a published blog post.
The Challenge of Formalizing Mathematics
A formal proof constitutes a sequence of arguments that mathematically verifies a specific hypothesis. The theorem Anthropic addressed is Fermat’s Last Theorem (FLT), an idea first proposed in 1637 concerning properties of positive whole numbers. The official proof for FLT was developed in 1995 by mathematician Andrew Wiles. This original work spans 129 pages and required months of rigorous verification.
Anthropic’s research focused on formalizing Wiles’ proof, which involves translating it into a format that can be automatically checked by computers. This process is valuable because it eliminates the possibility of human error and facilitates the sharing of mathematical knowledge among researchers.
A formalized proof is structured as code written in the Lean programming language, a specialized syntax used by mathematicians for hypothesis verification. The resulting proof generated by Anthropic contains 13 million lines of Lean code, making it the largest such file recorded.
Technical Hurdles and AI Breakthrough
Formalization is inherently difficult because proofs often lack detailed explanations necessary for computer comprehension, requiring manual input from Lean developers. Furthermore, the arguments within a proof are highly interdependent; consequently, a single error in the code can invalidate all subsequent lines. Previously, experts anticipated that formalizing Wiles’ proof would require several years of work.
According to Anthropic, the research team managed to complete the task in just 11 days using an internal research model. This algorithm was described as being roughly comparable to Claude Fable 5.1, the model’s immediate predecessor to GPT-6 Astra. The model achieved this efficiency by requiring only a limited amount of high-level guidance from human researchers. During the process, the system activated several dozen agents, generating 6 billion tokens of output and successfully proving no fewer than 29,500 intermediate theorems.
Anthropic noted that their initial attempts to formalize the proof were unsuccessful. The breakthrough occurred after the company granted Claude access to an open-source tool called Prove2Me. This software significantly aids AI agents by helping them determine the most optimal next step within a lengthy processing workflow, while also assisting in reducing computational inference costs.
Kevin Buzzard, a mathematician whose work Claude used to generate the formalized proof, commented on the achievement: “We see autoformalization of algebra, harmonic analysis, geometry and number theory, and we learn that AI autoformalization artefacts are now robust enough to be built upon; the proof is multi-layered.”
Industry Context and Future Research
This milestone follows a month-long period during which Anthropic used Claude to make another significant mathematical discovery—extracting new information about the Riemann zeta function, a central component of the highly challenging Riemann hypothesis.
Meanwhile, rival OpenAI Group PBC is also advancing mathematical research using its large language models. Last month, the company utilized its Astra model to solve several problems posed by Erdos and narrow down various open questions in the field of theoretical computer science.
