Claude completes Fermat proof in Lean, turning Fermat’s Last Theorem into a machine-checked milestone

A proof machines can check

Claude has completed a machine-checked Lean proof of Fermat’s Last Theorem, according to Anthropic’s September 4 announcement, which says the model worked “largely autonomously” for 11 days to produce the first complete computer-checked proof of the theorem [1]. The achievement is not that Claude discovered a new proof of Fermat’s Last Theorem, but that it formalized the existing mathematical route into a form a proof assistant can verify line by line [1].

That distinction matters. Fermat’s Last Theorem says that no positive integers (a), (b), and (c) satisfy (a^n + b^n = c^n) for any integer exponent (n > 2), a statement associated with Pierre de Fermat’s famous 17th-century marginal note [1]. Andrew Wiles proved the theorem in the 1990s, with the final published proof appearing in 1995 after a gap in the original 1993 announcement had to be repaired [1]. Claude’s role was to turn that human mathematical edifice into a formal artifact that Lean’s kernel can check.

Anthropic says Claude produced about 13 million lines of Lean code and proved 30,300 computer-verifiable theorems along the way, with 29,500 used in the final proof [1]. The company says the finished formalization uses only Lean’s three standard axioms, and that a comparator confirmed the statement matches Mathlib’s version of Fermat’s Last Theorem [1]. A public GitHub repository associated with the project describes the result as a complete machine-checked proof in Lean 4, built on Mathlib, with a final theorem stating the natural-number version of Fermat’s Last Theorem [1].

Why this is not “AI solved Fermat”

The headline is dramatic, but the careful reading is more important: Claude did not solve an open mathematical problem. Wiles already did that. What Claude appears to have done is solve a verification and translation problem at unprecedented scale [2].

A human proof is written for expert readers. It can rely on shared background, omit routine steps and point to existing literature. A formal proof cannot do that. Lean requires definitions, dependencies and inference steps to be explicit enough for a small checking kernel to accept them [1]. In that sense, Claude did not replace Wiles; it acted as an enormous formalization engine.

Tech Times, summarizing the same announcement, underlined this distinction: the result is a formalization of Wiles’s proof, not a new mathematical discovery [2]. That framing is essential because it prevents the story from becoming a misleading claim that a chatbot “found” Fermat’s proof. The accomplishment is different, but still major: an AI system helped convert one of the most famous proofs in mathematics into a binary, auditable software object.

Kevin Buzzard, the Imperial College London mathematician who had been leading a multi-year community effort to formalize Fermat’s Last Theorem, reviewed the result and said it proves the theorem with no assumptions beyond the axioms of mathematics, according to Anthropic’s post [1]. Anthropic also says Claude’s proof follows a simplified version of Wiles’s proof from Henri Darmon, Fred Diamond and Richard Taylor [1].

The 11-day run

The speed is what made the announcement feel like a rupture. Anthropic says the formalization of Fermat’s Last Theorem had been expected to take years; even the project blueprint for the initial phase ran to 86 pages [1]. Claude completed the run in 11 days, according to the company, after dozens of agents collaborated on definitions, intermediate theorems and increasingly difficult proof goals [1].

A 36Kr Europe report emphasized the same scale: 13 million lines of code, more than 29,000 supporting theorems and around 6 billion output tokens [3]. Silicon Report also reported that the multi-agent system consumed approximately 6 billion output tokens from an internal research model and that the proof was checked by Lean using only its standard axioms [4].

The project was linked to Tianyi Peng, an Anthropic researcher whose group at Columbia University works on AI formalization tools [1]. 36Kr highlighted Peng’s background in Tsinghua University’s Yao Class and described him as the team leader behind the effort [3]. Anthropic’s own account says Peng’s human input during the run was limited to occasional high-level instructions, such as prioritizing work around Jacobians or Mazur’s theorem [1].

Prove2Me as the scaffolding

The model was not simply asked to “prove Fermat” in one prompt. Anthropic says early attempts failed because agents lost track of the project state and stopped collaborating effectively [1]. That failure mode is familiar in long-horizon AI work: once a project becomes too large for any single context window, agents duplicate work, miss dependencies or lose sight of what remains unresolved.

The decisive shift came from Prove2Me, an open collaborative platform for formalizing mathematics designed by Peng and collaborators at Columbia University [1]. Anthropic says Prove2Me maintained a directed acyclic graph of theorem statements, separated theorem statements from proofs to speed compilation and kept natural-language descriptions so agents could search and reuse prior results [1].

Silicon Report described the system as a Claude Code-based multi-agent harness coordinated through Prove2Me, with dozens of agents defining concepts and assembling progressively harder statements [4]. Stack Futures reported that the run started in early August and crossed the finish line at 02:00:57 UTC on August 18, after an 11-day campaign [5].

This architecture is the technical heart of the story. The milestone is not only about a stronger model; it is also about external memory, project management and verification infrastructure. In other words, formal mathematics may become a domain where the winner is not the largest model alone, but the best combination of model, proof assistant and collaborative state system.

Why Lean changes the trust equation

Lean is important because it changes what “proof” means operationally. A conventional mathematical proof is trusted after expert review. A Lean proof is accepted only if it passes the proof assistant’s mechanical checks. That does not make all philosophical or software concerns disappear, but it does move the confidence problem from subjective expert judgment toward auditable computation.

Anthropic says the final proof was checked by Lean and that a comparator verified the theorem statement against Mathlib [1]. The associated repository documentation says a second independent Lean kernel implementation, nanoda, accepted an export of the same environment, adding another layer of checking [1]. That is a strong signal for formal verification because independent checking reduces the risk that the result depends on one implementation path.

Still, the caveats are real. A proof assistant verifies the formal statement it is given; it does not decide whether every name is intuitive or whether a human exposition is enlightening. Anthropic explicitly argues that formalized proofs should not replace human-readable mathematical explanations, even as they may become necessary for keeping up with AI-generated mathematical output [1].

The market and community reaction

Reports around the result also triggered a sharp reaction outside academic mathematics. One report said a prediction market tied to the Fermat formalization jumped to 99% after Claude’s Lean proof appeared [6]. That market movement should not be treated as a mathematical validation in itself, but it shows how quickly the announcement was interpreted as a credible milestone by observers tracking AI and formal verification.

The reaction from mathematics and AI communities has been more nuanced. The proof is machine-checkable, but it is also huge: Anthropic says it is more than five times the size of Mathlib, and notes that the proof is likely much longer than necessary because Mathlib is concise and heavily reviewed [1]. A 13-million-line proof is not a polished textbook argument. It is closer to an industrial-scale verification artifact.

That matters for what comes next. If AI systems can routinely produce vast, correct but unwieldy formalizations, the next challenge will be compression, readability and integration into community libraries. Mathematicians will want shorter proofs, clearer dependency graphs and reusable components rather than one-off mountains of code.

A marker for AI labs

For AI labs, the strategic implication is obvious. Formal mathematics offers a rare testbed where correctness can be binary: either the proof checks or it does not. That makes it attractive for evaluating frontier models in domains where mistakes are costly and auditability is valuable.

Anthropic argues that automatic formalization could help root out errors in the mathematical corpus and reduce the burden on referees [1]. Silicon Report noted Anthropic’s additional claim that three personal Claude Max plans were used in a separate experiment to formalize Vinogradov’s Three Primes Theorem in three days, suggesting that smaller-scale collaborative formalization may become more widely accessible [4].

The immediate result is therefore both narrow and large. Narrow, because Fermat’s Last Theorem was already proved. Large, because one of the most celebrated proofs in mathematics has now become a machine-checkable object through an AI-driven workflow. If the result holds up under continued scrutiny, Claude’s Fermat run will be remembered less as a replacement for mathematicians than as a sign that formal verification is entering its acceleration phase.

Sources from the last 72 hours

1. [1]Formalizing Fermat's Last TheoremSep 4, 2026, 12:00 AM UTC
2. [2]Fermat's Last Theorem Machine-Checked: Claude Completes in 11 Days What Took Years to PlanSep 5, 2026, 1:39 PM UTC
3. [3]Claude Proves Fermat's Last Theorem for the First Time: Tsinghua University Yao Class Top Talents Step Into the Groundbreaking Mathematical BreakthroughSep 5, 2026, 1:16 AM UTC
4. [4]Anthropic says Claude formalized Fermat's Last Theorem proof in 11 daysSep 5, 2026, 7:00 AM UTC
5. [5]Claude Formalized Fermat's Last Theorem in 11 Days — 13 Million Lines of Lean, 29,500 ProofsSep 4, 2026, 12:00 AM UTC
6. [6]Fermat’s Last Theorem Prediction Market Spikes to 99% After Claude’s Lean ProofSep 5, 2026, 12:00 AM UTC

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