Anthropic Claude Completes First Machine-Checked Proof of Fermat's Last Theorem in Lean

Milestone Formalization of a 350-Year-Old Problem

Anthropic researchers announced on September 4, 2026, the completion of the first end-to-end, computer-checked formal proof of Fermat's Last Theorem (FLT). The formalization translates the human mathematical reasoning behind the historic conjecture into machine-verifiable code using the Lean programming language. Pierre de Fermat originally jotted down the conjecture around 1637, positing that no three positive integers $a, b, c$ satisfy $a^n + b^n = c^n$ for any integer $n > 2$. Sir Andrew Wiles published the first recognized mathematical proof in 1995 across 129 pages, an achievement that took human experts months of intense scrutiny to evaluate and accept ^[2][8].

Converting Wiles's proof into formal logic—where every deduction is step-by-step verifiable by software—has stood as one of the most formidable targets in computational mathematics since Dutch computer scientist Jan Bergstra proposed formalizing it a decade after Wiles's publication. The AI system Claude carried out the extensive formalization work over an 11-day period under the direction of Tianyi Peng, an Anthropic researcher whose research group at Columbia University creates tools for automated formalization. Anthropic published the resulting proof files in an open public repository under the identifier `anthropics/fermats-last-theorem` ^[2][8].

Independent Compilation and Verification by Kevin Buzzard

Crucially for the broader scientific community, the validity of the codebase does not rely on Anthropic's self-reporting. Kevin Buzzard, a professor of pure mathematics at Imperial College London who has led an ongoing multi-year community effort to formalize Fermat's Last Theorem in Lean, independently audited the released codebase. Buzzard downloaded the public repository, compiled the proof directly on his own hardware, ran the Lean kernel checker, and confirmed that the mathematical argument holds completely ^[2][3][8].

Buzzard verified that the proof is entirely "sorry-free"—meaning it contains zero unproven placeholders, gaps, or conjectures—and matches Mathlib's formal definition of Fermat's Last Theorem while resting exclusively on Lean's three standard mathematical axioms (`propext`, `Classical.choice`, and `Quot.sound`). Conceding the formalization race in a post titled "FLT: Anthropic has beaten me to it," Buzzard affirmed that the work formalizes complex components spanning algebra, harmonic analysis, geometry, and number theory. He noted that the artifacts produced by automated formalization have proven sufficiently robust and multi-layered to support intricate subsequent deductions ^[2][3].

Architecture, Scale, and Foundational Dependencies

The formalization of Fermat's Last Theorem represents a massive engineering and logical undertaking. According to Anthropic and tracking analyses, the completed proof chain required Claude to generate approximately 13 million lines of Lean code, establishing more than 29,500 intermediate theorems and 1,450 formal definitions, all checked through the Lean kernel. The work follows the modern Frey-curve and modularity-lifting architecture established by Wiles and Richard Taylor, rather than classical approaches like Ernst Kummer's regular-prime technique, which cannot address the complete theorem ^[2][8].

Anthropic built this formalization on top of foundational open infrastructure developed over years by the interactive theorem-proving community. This foundation includes Lean's standard library Mathlib, the Imperial College FLT-in-Lean project, and existing formalizations such as `flt-regular`. Specifically, the proof addresses the small prime cases ($p < 17$) by linking to `flt-regular`, which fully covers all cases below 37 (the smallest irregular prime). Anthropic disclosed this dependency openly. In doing so, the completed proof also marks the formal conclusion of Freek Wiedijk's 20-year "100 theorems" challenge list, completing a long-standing historical benchmark in formal logic ^[3][8].

Distinguishing Mathematical Output From Process Claims

In analyzing this development, observers have emphasized the importance of distinguishing between the verified mathematical artifact and Anthropic's process descriptions. The existence of a verified, sorry-free Lean proof resting on standard axioms is externally and adversarially confirmed by the leader of the competing academic project. Anyone with access to the public Lean repository can independently run the Lean kernel to check every step ^[3].

In contrast, Anthropic's account of how the artifact was generated—specifically the claims of completing the project in roughly 11 days, consuming approximately six billion output tokens, orchestrating dozens of agents through a proprietary "Prove2Me" layer, and operating largely autonomously with limited human steering—remains self-reported by the company. The verified code repository confirms that the output is syntactically sound and logically complete Lean, but it does not independently document the exact degree of human intervention or compute resources deployed behind the scenes ^[3].

Implications for Mathematics and Automated Verification

Unlike experimental AI research that attempts to generate novel mathematical conjectures, this accomplishment centers entirely on formal verification. Translating an informal 129-page human proof into machine-verifiable logic eliminates human ambiguity and guarantees that no subtle oversights remain hidden in the reasoning chain. Because human peer review of complex proofs routinely requires months or years, the ability to automate rigorous formalization offers a scalable safeguard as mathematical research expands ^[2][8].

While human mathematicians laid years of groundwork to establish the required libraries and frameworks, this project demonstrates that AI systems can navigate dense logical dependencies without losing coherence across thousands of nested theorems. For computer science and pure mathematics, the transition from multi-year manual formalization projects toward automated, machine-checked verification signals a major shift in how the global academic community may validate future knowledge ^[2][3][8].

Sources

1. Formalizing Fermat's Last Theorem \ Anthropic www.anthropic.com
2. Anthropic and Claude Formalized Fermat's Last Theorem — and the Rival Project's Leader Compiled the Proof Himself - FourWeekMBA fourweekmba.com
3. Claude helps complete first formalized proof of Fermat's Last Theorem cryptobriefing.com