Anthropic Says Claude Wrote First Machine-Checked Proof of Fermat's Last Theorem: 13-Million-Line Lean 4 Breakthrough

In what mathematicians are calling one of the most profound milestones in the history of computational science, Anthropic announced on September 5, 2026, that Claude has completed the world's first fully formalized, machine-checked Lean 4 proof of Fermat's Last Theorem.

Over the course of an intensive 11-day autonomous run in August, Claude generated a staggering 13-million-line proof file, translating Sir Andrew Wiles' monumental 1994 paper and its deep algebraic geometry foundations into mathematically infallible, machine-verified code.

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#1. The 350-Year Challenge: From Pierre de Fermat to Lean 4

First conjectured by French mathematician Pierre de Fermat in 1637—who famously jotted in the margin of his copy of Diophantus' Arithmetica that he had found a "truly marvelous proof which this margin is too narrow to contain"—the theorem asserts that:

$$\text{No three positive integers } a, b, c \text{ can satisfy } a^n + b^n = c^n \text{ for any integer value of } n > 2.$$

While Andrew Wiles famously proved the theorem in 1994 using the Modularity Theorem for semistable elliptic curves, human proofs inevitably span hundreds of pages of intricate prose where subtle assumptions can hide unnoticed.

mermaidCopy
graph TD
    A[Andrew Wiles 1994 Informal Mathematical Proof] --> B[Anthropic Mathematical Decomposition Engine]
    B --> C[Claude Multi-Agent Proof Synthesis Cluster]
    
    C --> D[Elliptic Curves & Galois Representations]
    C --> E[Modular Forms & Ribet Theorem]
    C --> F[Taniyama-Shimura-Weil Conjecture]
    
    D --> G[Lean 4 Compiler & Kernel Verifier]
    E --> G
    F --> G
    
    G --> H[13 Million Lines of Machine-Checked Formal Proof]

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#2. Technical Feat: How Claude Wrote 13 Million Lines of Verified Logic

The formalization was executed using Lean 4, an interactive theorem prover and functional programming language maintained by the Lean FRO.

Unlike standard natural language output where an AI might hallucinate convincing mathematical nonsense, the Lean 4 compiler is merciless: if a single lemma, type definition, or inductive step is flawed, the entire file fails to compile.

#Key Algorithmic Innovations in Claude's Run:

1. Backtracking Proof Search: When encountering a syntactic or logical impasse in elliptic curve invariants, Claude systematically branched out alternative proof paths, pruning circular reasoning chains.
2. Automated Library Expansion: Claude generated tens of thousands of missing intermediate lemmas for Mathlib (Lean's open-source mathematical library), substantially expanding the global foundation of computer-verified mathematics.
3. Self-Correcting Compiler Loops: Claude consumed compiler error logs directly from the Lean 4 language server, adapting its tactics dynamically until the compiler emitted a flawless exit code.

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#3. Why Machine Verification Changes Modern Science

For centuries, the gold standard of scientific truth was peer review—a subjective process vulnerable to fatigue and human oversight.

By demonstrating that AI can autonomously translate hyper-abstract theoretical mathematics into machine-verified logic:

• Flawless Software Verification: The exact same formal proof techniques can now be applied to mission-critical software systems, including cryptographic protocols, flight control computers, and smart contract architectures.
• Acceleration of Unsolved Conjectures: Claude's breakthrough suggests that long-standing open mathematical problems—such as the Riemann Hypothesis or P vs NP—may soon benefit from AI-led formal exploration.

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