Anthropic's Claude Autonomously Formalizes Fermat's Last Theorem in Lean

In a landmark milestone for artificial intelligence and computer-assisted mathematics, Anthropic announced that its Claude model has completed the first end-to-end, machine-verified proof of Fermat’s Last Theorem (FLT). Working largely autonomously over the course of just 11 days, an ensemble of AI agents orchestrated via the Prove2Me platform generated 13 million lines of code written in the Lean programming language. Along the way, the system successfully verified 30,300 theorems, with 29,511 woven directly into the final verified proof structure, according to Anthropic's research announcement.

Background

First conjectured in 1637 by French mathematician Pierre de Fermat, Fermat’s Last Theorem posits that no three positive integers $a$, $b$, and $c$ can satisfy the equation $a^n + b^n = c^n$ for any integer value of $n$ greater than 2. For more than 350 years, the assertion stood as one of the most stubborn and celebrated open questions in mathematics.

In 1995, British mathematician Sir Andrew Wiles finally proved the conjecture in a monumental 129-page breakthrough. However, verifying Wiles's original proof required months of painstaking peer review by leading specialists because human mathematical prose routinely omits intermediate steps deemed "obvious" to experts.

To eliminate human ambiguity and potential logical gaps, the mathematical community increasingly relies on "formal proof assistants" like Lean. In formalization, mathematical arguments are translated into rigorous code where every single logical step is mechanically checked back to foundational axioms. While powerful, formalizing modern mathematics by hand is notoriously grueling. In 2024, Kevin Buzzard at Imperial College London initiated a major community effort to formalize FLT in Lean—a complex endeavor that researchers anticipated would require several years of dedicated human labor.

What Happened

Rather than taking years, the formalization was accomplished in under two weeks. The project was led by Anthropic researcher Tianyi Peng, whose group at Columbia University developed Prove2Me, an open collaborative platform designed to track, coordinate, and host scalable formal proofs, as outlined in an arXiv research paper.

Operating through a multi-agent harness driven by an internal research model comparable to Claude Fable 5.1, dozens of Claude agents collaborated continuously. The system followed an established exposition of Wiles’s work by Henri Darmon, Fred Diamond, and Richard Taylor, working through intricate concepts such as Frey curves, Ribet’s theorem, and Taylor–Wiles patching.

The AI system broke down massive mathematical structures into discrete tasks, establishing definitions, proving intermediate lemmas, and assembling the dependency graph. Over 11 days of autonomous execution, the run consumed approximately six billion output tokens and expanded to 13 million lines of Lean code—more than five times the size of Mathlib, the primary mathematical library for Lean.

Human intervention remained minimal, limited to high-level steerage from Peng to prioritize specific mathematical sub-problems, such as emphasizing Mazur’s theorem or prioritizing scheme-theoretic machinery. The completed proof was fully compiled and validated by Lean's standard kernel using only its three fundamental axioms, with Lean tools confirming the overarching statement precisely matches Mathlib’s canonical definition of Fermat’s Last Theorem.

Why It Matters

This achievement represents a qualitative shift in automated reasoning and the future of science:

• Collapsing the Autoformalization Cost Curve: Translating advanced mathematical literature into verifiable software has historically represented a severe bottleneck. Demonstrating that AI can independently generate millions of lines of syntactically and logically sound Lean code suggests the timeline for digitizing modern mathematics has compressed from decades to months.
• Eliminating AI Hallucinations via Formal Verification: Large language models are notorious for producing persuasive yet subtly flawed arguments. By pairing generative AI with a strict, non-negotiable formal verification kernel like Lean, the system achieves an absolute standard of correctness. If the Lean compiler accepts the code, the proof contains no logical leaps.
• Scalable Multi-Agent Collaboration: The run provides a robust template for autonomous AI collaboration. Rather than relying on a single context window, dozens of Claude agents coordinated via the Prove2Me dependency graph to independently claim sub-problems, verify lemmas, and unblock downstream tasks without collapsing into uncoordinated loops.

What's Next

While the Lean kernel guarantees internal logical consistency, significant work remains. Independent observers and domain experts emphasize a "semantic review gap": researchers must still verify that Claude’s intermediate machine-generated definitions authentically correspond to the mathematical objects intended by human mathematicians, rather than degenerate shortcuts, as noted by Superpower Daily.

Furthermore, the sheer volume of output—13 million lines containing boilerplate and local lemmas—is far too unrefined for immediate integration into Mathlib. The next major challenge for AI labs will be proof refactoring, automated compression, and idiomatic synthesis so that AI-generated mathematics can be easily studied and maintained by humans.

Beyond pure mathematics, the success of this run signals broader implications for safety-critical domains. Techniques refined during this 11-day experiment will likely find immediate applications in formal hardware verification, cryptographic protocol auditing, and mission-critical software engineering, where provable correctness is just as vital as it is in the margins of Fermat's notebook.