Anthropic's AI model, Claude, has autonomously formalized Fermat's Last Theorem, producing a computer-checked proof in just 11 days. The proof, consisting of 13 million lines of Lean code, marks a significant milestone in AI-assisted mathematics.
TL;DR
- Claude autonomously formalized Fermat's Last Theorem in 11 days, producing a 13M-line Lean proof.
- This achievement could reduce the burden of formal verification and increase trust in mathematical knowledge.
- Anthropic is making Claude more accessible to external researchers, including mathematicians.
What happened
Anthropic's AI model, Claude, has successfully formalized Fermat's Last Theorem, producing a complete, computer-checked proof in just 11 days. The proof, written in the Lean programming language, consists of 13 million lines of code and proves 29,500 intermediate theorems.
This achievement builds upon a decade of work by mathematicians to formalize Wiles's 1995 proof of the theorem. The formalization process was expected to take years, but Claude completed it in a fraction of that time.
The proof was reviewed by Kevin Buzzard, a mathematician at Imperial College London, who confirmed its correctness. Buzzard noted that the proof is multi-layered and robust, covering a wide range of mathematical disciplines.
Why it matters
This achievement demonstrates the potential of AI to assist in formalizing complex mathematical proofs, reducing the burden of verification and increasing trust in mathematical knowledge.
As AI produces ever more proofs, the ability to easily formalize work can lighten the burden of evaluating new results, which can take years. This could be particularly valuable for startups and researchers working on AI-driven mathematical discoveries.
Anthropic is making Claude more accessible to external researchers, including mathematicians, through free and discounted subscriptions and research credits. This could enable more collaborative formalization efforts and accelerate progress in mathematics.
Key facts
- Claude formalized Fermat's Last Theorem in 11 days, producing a 13M-line Lean proof.
- The proof includes 29,500 intermediate theorems.
- The proof was reviewed and confirmed by Kevin Buzzard, a mathematician at Imperial College London.
- Anthropic is offering free and discounted subscriptions and research credits to external researchers, including mathematicians.
- The formalization process was expected to take years, but Claude completed it in a fraction of that time.
- The proof follows a simplified version of Wiles's proof from Darmon, Diamond and Taylor.
- The effort succeeded using Prove2Me, an open collaborative platform for formalizing mathematics.
- The finished proof was checked by Lean and uses just Lean's three standard axioms.
Context
Fermat's Last Theorem is one of the most famous mathematical conjectures of all time. It states that no positive integers a, b, c satisfy aⁿ + bⁿ = cⁿ for any n > 2.
The first proof of the theorem was published by Sir Andrew Wiles in 1995, after more than 350 years of mathematical effort. The proof is highly complex and relies on modern mathematical techniques.
Formalizing a mathematical proof involves converting the reasoning into a form that computers can check automatically. This process can be time-consuming and labor-intensive, but it ensures the correctness of the proof beyond a doubt.
