Claude Completes a Computer-Checked Formalization of Fermat’s Last Theorem in Just 11 Days
Anthropic says Claude has completed the first end-to-end, computer-checked formalization of Fermat’s Last Theorem, turning one of mathematics’ most famous proofs into a form that the Lean proof assistant can verify automatically. The company announced the achievement on September 4, 2026, saying Claude worked largely autonomously for 11 days and produced roughly 13 million lines of Lean code.
The result is not a new proof of Fermat’s Last Theorem. The theorem was proved by mathematician Andrew Wiles in the 1990s. Instead, the breakthrough is that an AI system helped translate the mathematical argument into a formal language in which a computer can check every required step.
That distinction is important. Mathematical proofs written for humans routinely omit intermediate steps that experts consider obvious. Formal proof systems such as Lean do not allow those gaps. Every definition, inference and dependency has to be expressed precisely enough for the computer to verify.
Anthropic says Claude proved approximately 29,500 intermediate theorems while building the final formalization. The company argues that the achievement points toward a future in which AI could dramatically accelerate the process of turning existing mathematical knowledge into machine-checkable form.
What Is Fermat’s Last Theorem?
Fermat’s Last Theorem states that there are no positive integers a, b and c that satisfy the equation an + bn = cn when n is an integer greater than 2.
The statement itself is remarkably simple. The proof is not.
Pierre de Fermat wrote the famous claim in the margin of a copy of an ancient mathematical text around 1637, suggesting that he had found a proof but that the margin was too small to contain it. The problem remained unresolved for centuries.
Andrew Wiles finally proved the theorem in the 1990s using advanced mathematics involving elliptic curves, modular forms and other areas of number theory. The resulting argument was extremely sophisticated and required substantial mathematical knowledge to understand and verify.
For decades after Wiles’s proof, mathematicians continued working on ways to formalize major mathematical results so that proof assistants could independently check them.
What Does “Formalizing” a Proof Mean?
A conventional mathematical proof is written for human readers. Authors can leave out routine steps, refer to established results and rely on shared mathematical intuition.
A formal proof is different. A proof assistant such as Lean requires mathematical statements and their logical dependencies to be expressed in a precise programming language. The system then checks whether the supplied sequence of deductions follows from its foundations.
This creates a powerful form of verification. Instead of asking a human mathematician to inspect thousands of pages and decide whether every argument is valid, researchers can ask a computer proof system to check the formalized result.
The difficulty is that formalization can be extraordinarily time-consuming. A human-readable proof may contain sentences that summarize dozens or hundreds of logical steps. Translating those steps into Lean can require substantial mathematical and programming work.
That is the problem Anthropic says Claude has helped solve at an unusually large scale.
Claude Worked for 11 Days
Anthropic researcher Tianyi Peng and collaborators set out to test whether Claude could make meaningful progress on formalizing Fermat’s Last Theorem.
According to Anthropic, the experiment went much further than expected. Claude agents worked for 11 days, largely autonomously, defining mathematical concepts, proving intermediate results and assembling those results into the final formal proof.
Anthropic reports that the system produced 13 million lines of Lean code. Along the way, it established computer-checked proofs for about 30,300 theorems, with approximately 29,500 of them ultimately used in the final proof.
The scale is striking. Anthropic says the resulting proof is more than five times the size of Mathlib, the major community library of formalized mathematics on which the project builds.
The achievement also involved multiple Claude agents collaborating on different portions of the work. Rather than relying on one uninterrupted generation, the system could divide the enormous formalization task into smaller mathematical problems and build progressively toward the complete result.
This Was Not an AI Discovery of a New Proof
One of the most important points for understanding the announcement is that Claude did not independently discover a previously unknown proof of Fermat’s Last Theorem.
The underlying mathematical result was already established by Wiles and other mathematicians. Anthropic’s achievement is better described as AI-assisted autoformalization: converting an existing human mathematical argument into a machine-verifiable formal proof.
That may sound less dramatic than discovering a new theorem, but it addresses a major problem in modern mathematics.
As mathematical knowledge grows more complicated, verifying new work can require enormous amounts of specialist effort. A formal system can provide a much more mechanically rigorous method of checking whether a proof follows from its stated foundations.
AI that can automate part of this process could therefore become a powerful research assistant even when it is not generating fundamentally new mathematics.
Why the Lean Proof Assistant Matters
Lean is a computer-assisted proof system designed to let mathematicians express formal mathematics in a language that computers can verify.
The system checks whether statements follow according to its formal rules. This means that a completed Lean proof is not simply an AI-generated explanation that sounds convincing. It is an executable formal artifact that can be checked by the proof assistant.
That distinction becomes increasingly important as AI systems become better at producing persuasive mathematical prose.
A language model can generate a proof that appears plausible while containing a subtle logical error. A formal proof assistant provides an additional layer of rigor because the machine will reject an invalid formal step.
AI and proof assistants therefore complement each other. The AI can perform much of the difficult translation and construction work, while the formal system provides deterministic verification of the resulting artifact.
A Major Step for AI-Assisted Mathematics
Anthropic believes the Fermat project demonstrates that AI can now tackle formalization tasks that were previously expected to take years.
The implications extend beyond one famous theorem.
If AI systems can reliably translate sophisticated mathematical arguments into formal languages, researchers could eventually use them to formalize large portions of the mathematical literature. That could make existing results easier to verify, reuse and connect.
It could also help identify gaps or inconsistencies in complicated mathematical arguments. A proof that has been successfully formalized has to make its assumptions and logical dependencies explicit.
This could become especially useful for areas of mathematics where proofs depend on long chains of previous results.
The Human Role Is Still Important
The achievement does not mean mathematicians are about to become unnecessary.
Formalization remains a highly specialized task, and humans still need to choose appropriate mathematical strategies, understand the underlying arguments and decide how a proof should be represented.
Anthropic says Claude worked with high-level guidance rather than operating completely without human involvement. Researchers also had to establish the project’s direction and provide the environment in which the AI could work.
There is another limitation: a massive formal proof may be difficult for humans to read even if it is easy for a computer to verify.
Claude’s 13 million lines of Lean code are not equivalent to a readable mathematical exposition. The value comes partly from the fact that the proof can be checked mechanically, not because a human mathematician would want to read every line.
Could AI Formalize More of Mathematics?
The most interesting question now is what comes next.
Fermat’s Last Theorem is an unusually ambitious target, but mathematics contains thousands of important results that have not yet been fully formalized.
If AI systems can automate increasingly large portions of this process, formalization could become much faster and less expensive.
Researchers could eventually ask AI systems to convert papers into formal proofs, connect results from different mathematical fields and create reusable libraries of machine-verified knowledge.
This could also change how future mathematical research is published. Instead of sharing only a human-readable proof, researchers might increasingly publish a formal proof that can be checked automatically alongside the explanatory version.
The Biggest Challenge May Be Scale
Claude’s success also highlights a potential problem: formal proofs can become enormous.
A 13-million-line artifact is impressive, but maintaining, updating and understanding such a large body of formal code can create its own engineering challenges.
Researchers will need reliable tools for navigating formal proofs, reusing existing theorems and checking whether future changes break earlier results.
The mathematical community will also need standards for evaluating AI-generated formalizations, documenting the models and computational resources involved, and making results reproducible.
In other words, AI may solve part of the formalization bottleneck while creating a new infrastructure challenge around managing machine-generated mathematics.
What This Means for AI Research
The Fermat formalization is another example of AI moving beyond conventional chatbots.
Advanced models are increasingly being tested as autonomous research systems capable of planning tasks, writing code, using specialized tools and working for extended periods.
Mathematics is particularly well suited to this approach because formal proof systems provide an objective feedback mechanism. The AI either produces code that the proof assistant accepts or it does not.
That creates a valuable environment for agentic AI research. Instead of relying entirely on human judgments about whether an answer looks correct, AI systems can receive machine-generated feedback and repeatedly improve their attempts.
The same general approach could potentially be applied to software verification, scientific computing and other technical domains where results can be checked automatically.
What Happens Next?
Anthropic’s result is likely to encourage more experiments combining frontier AI models with formal mathematics systems.
The next milestones may involve formalizing additional difficult theorems, reducing the amount of human guidance required, improving the readability of AI-generated proofs and lowering the computational cost of producing them.
Researchers may also investigate whether AI can use formalization not just to verify known mathematics, but to discover genuinely new results through a cycle of conjecture, proof generation and machine verification.
That would represent a substantially different milestone. Instead of translating an existing proof, an AI system would propose a new mathematical statement and establish its validity.
For now, Anthropic’s Fermat result is best understood as a major demonstration of AI-assisted verification rather than autonomous mathematical discovery.
Conclusion
Claude’s completion of a computer-checked formalization of Fermat’s Last Theorem is a significant milestone in the relationship between artificial intelligence and mathematics.
The theorem itself was solved by humans decades ago. What is new is the ability of an AI system to take on the enormous task of translating that sophisticated mathematics into a form that a computer can verify.
Working largely autonomously for 11 days, Claude generated millions of lines of Lean code and thousands of intermediate theorems to construct the formalization. The result demonstrates that AI can now participate in mathematical work at a scale that would have been extremely difficult to manage manually.
The broader significance may ultimately be even greater than the theorem itself. If AI can help formalize increasingly large parts of mathematics, the technology could make rigorous verification faster, strengthen the reliability of mathematical knowledge and create new tools for researchers.
The next frontier is not simply asking whether AI can solve mathematics. It is asking whether AI can help make the entire mathematical knowledge base easier for machines—and humans—to verify.
FAQ
Did Claude discover Fermat’s Last Theorem?
No. Fermat’s Last Theorem was proved by Andrew Wiles in the 1990s. Claude’s achievement was to produce a complete computer-checked formalization of the existing mathematical result.
What did Claude actually produce?
Anthropic says Claude produced approximately 13 million lines of Lean code and proved about 29,500 intermediate theorems used in the final formalization.
What is Lean?
Lean is a formal proof assistant that allows mathematical statements and proofs to be expressed in a precise language that a computer can check.
Why is formalizing mathematics useful?
Formalization makes the logical steps of a proof explicit and allows a computer to verify them. This can reduce the risk of unnoticed errors and make complex mathematical results easier to check mechanically.
Did Claude work completely without humans?
Anthropic describes the work as largely autonomous, but researchers provided high-level direction and the technical environment. The project should therefore be viewed as AI-assisted formalization rather than a completely independent mathematical research process.
Could AI eventually discover new mathematics?
Potentially. Formal proof systems provide a way for AI-generated mathematical ideas to be tested rigorously. However, Claude’s Fermat achievement was a formalization of an existing proof, not the discovery of a new theorem or proof.
Sources
- Anthropic Research — “Formalizing Fermat’s Last Theorem,” published September 4, 2026.
- Anthropic Research — research overview and related work on Claude’s mathematical capabilities.
- Hacker News — community discussion of Anthropic’s Fermat formalization and its relationship to existing Lean formalization efforts.
- Crypto Briefing — independent reporting on Claude’s machine-checked Fermat formalization.
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