Claude Formalizes Fermat's Last Theorem in 11 Days

Anthropic announced on September 4, 2026, that Claude had generated a complete computer-checked proof of Fermat’s Last Theorem. The work took 11 days and was carried out largely autonomously. It was written in Lean, totaling 13 million lines. The number of intermediate theorems proved reached 29,500. Reporting by jlebar on Hacker News (Best) describes the achievement as the first end-to-end computer-checked proof.

According to Anthropic’s official announcement, establishing verifiability itself is the core of this effort. Rather than a new mathematical discovery, the value lies in converting the existing proof into a machine-checkable form. The company describes the effort as a step toward verifiability for mathematics as a whole.

We are sharing the first complete computer-checked proof of Fermat’s Last Theorem.

The above is the opening sentence of the official announcement. The claim Pierre de Fermat wrote in a margin around 1637 became the subject of autoformalization nearly 360 years later. Andrew Wiles’s 129-page proof in 1995 took months to verify. This can be seen as an attempt to bear that weight through machine verification.

How the Proof Was Completed Autonomously

in 11 Days

The project was led by Anthropic researcher Tianyi Peng. He leads a group at Columbia University developing tools for AI formalization. The initial aim was to test whether Claude could make progress on formalizing Fermat’s Last Theorem. The result exceeded expectations, yielding an end-to-end proof in 11 days. The work is said to have proceeded largely autonomously, with limited human intervention.

The scale of the output differs by orders of magnitude from previous manual formalization. The Lean code spans 13 million lines, with 29,500 intermediate theorems. Rather than a single massive proof, it is a multi-layered structure. It covers algebra, harmonic analysis, geometry, and number theory. Its distinctive feature is that the autoformalization output has a robustness that allows it to be built upon.

The completed proof was shared with Kevin Buzzard of Imperial College London. He has led a community formalization project since 2024. He assessed that the theorem had been proved with no assumptions other than the axioms of mathematics. He said autoformalization output has entered a stage where it can serve as a foundation for others to build on.

This extraordinary autoformalization achievement, which Anthropic researchers say only took 11 days, proves Fermat’s Last Theorem with no assumptions other than the axioms of mathematics.

Breakdown of the 13 Million Lines and

29,500 Theorems

A scale of 13 million lines is not a volume for humans to read directly. It is a volume for machines to check and reuse. The 29,500 intermediate theorems form a hierarchy that supports the higher-level logic. Each layer depends on lower-level definitions and lemmas. Because dependencies are explicit, any error can be detected mechanically.

The covered areas span algebra, harmonic analysis, geometry, and number theory. The proof of Fermat’s Last Theorem is not confined to a single field. It requires tools such as elliptic curves, modular forms, and Galois representations. The core task was encoding them into Lean’s logical system. Maintaining consistency of definitions across fields was apparently the challenge.

The multi-layered nature of the output is important for reuse. The lower-level lemmas could potentially be repurposed for proving other theorems. Formalized definitions leave no ambiguity. Subsequent research can build on the same foundation. This can be seen as marking a stage where autoformalization outputs become assets.

Details on computing resources and verification time are not included in the published information. The mechanism for checking 13 million lines in a practical amount of time is itself an issue. The performance of Lean’s checker and the design of split verification hold the key. Operational knowledge from large-scale formalization is expected to become a shared asset. Structural analysis of the published proof will be the next focus.

Formalization to Reduce the Burden of Proof

Verification

Verifying mathematical proofs carries a different burden from checking calculations. It involves assembling complex chains of logic, where a single break undermines the whole. Manual verification by peer review can take months to years. The verification process for Wiles’s first proof is an example. Formalization is a method of delegating this burden to machines.

The novelty here lies on the verification side. The recent high-profile AI effort around the Riemann hypothesis focused on generating new mathematics. In contrast, this work improved the checkability of a known theorem. It is the idea of checking proofs by machine, just as calculators check calculations. Its significance lies in the clear separation of generation and verification.

As AI generates more proofs, the burden of evaluation grows. If formalization becomes easier, mechanical checking before peer review could become standard. Early detection of errors and clarification of assumptions will advance. It will become easier to maintain trust in the body of mathematical knowledge. Developing verification infrastructure is seen as directly linked to faster research.

The push for greater reliability parallels AI infrastructure development in other fields. In computing on encrypted data, Google Announces HEIR for Encrypted Computation of AI Models aims to achieve both verifiability and confidentiality. In changing development environments, OpenAI to End AI Browser Atlas, Move Features to Chrome Extension and App shows the integration and reconfiguration of tools. In manufacturing demonstrations, Slate Achieves Cheapest EV with Chinese-made LFP Batteries tests the link between design and mass production. The formalization of mathematics can likewise be seen as entering a stage of implementation and operation.

Relationship to the Manual Formalization Project

The idea of formalization was proposed about 10 years after Wiles’s proof. Dutch computer scientist Jan Bergstra proposed converting it into a machine-checkable form. Development of methods to encode complex proofs then continued. In 2024, Buzzard launched a collaborative project to formalize it in Lean. The background assumes years of collaborative work.

Claude’s achievement does not replace this collaborative project. It likely builds on manually developed definitions, tactics, and operational knowledge. The 11-day period should be understood as execution time excluding groundwork. It is reasonable to see it as the result of advanced division of labor between automation and humans. The two can be seen as complementary rather than competing.

Lean has become a shared foundation for formalization as a proof assistant. Its strengths are rigorous definitions and a reliable checker. The community’s accumulated mathematics libraries supported automation. This output was also built on those assets. The result reflects the interaction between infrastructure and automation.

The focus will now shift to quality control of the output. How to maintain and update 13 million lines will become an issue. Keeping up with definition changes and managing dependencies will be needed. The allocation between human design decisions and machine workload will be questioned. The management of the collaborative project itself is expected to change.

Impact on the Trust Foundation of

Mathematical Research

The fact that autoformalization can be applied to complex theorems carries weight. The vision of a future where all of mathematics can be machine-checked has become more realistic. Proof assumptions and inference paths are made explicit. The cost of rechecking by third parties falls. Transparency of the knowledge system can be said to increase.

The division of labor in research will also change. Humans will be able to focus more on strategy and conceptual design. Machines will handle lemma expansion and consistency checks. Peer review will layer human judgment on top of machine checking. This is seen as a system that can achieve both productivity and rigor.

Spillover into education is also expected. Formalized proofs become step-by-step learning materials. Dependencies from definitions to theorems can be visualized. Sources of errors are easier to locate. They can be seen as teaching materials well suited to training rigorous thinking.

On the other hand, trust in the checker is a prerequisite. The correctness of Lean’s core checking logic supports the whole. The choice of axiom system and implementation errors remain subjects for scrutiny. The larger the output, the more important audit design becomes. How to explain the chain of technical trust remains an issue.

Editorial Opinion

In the short term, adoption of mechanical checking before peer review is expected to advance. Use of formalization in Lean may expand in collaborative mathematics research. Development of lemma collections as reusable proof components is expected to accelerate. Reducing the verification burden has reached a stage directly linked to research speed.

In the long term, structural change in the knowledge base of mathematics is expected to proceed. New theorems may build on multi-layered formalized assets. Use of rigorous reasoning is expected to spread in education and industrial applications. The locus of trust is seen as shifting from people to systems.

Can humans audit 13 million auto-generated lines? Where should the boundary between verifiability and understandability be drawn? On what conditions should trust be placed in machines? The answers to these questions seem likely to shape future operational design.

References

• “Formalizing Fermat’s Last Theorem”, by jlebar — Hacker News (Best), 2026-09-04T18:42:56.000Z (ARR)
• Source URL: https://www.anthropic.com/research/formalizing-fermats-last-theorem