Anthropic's Claude Produces the First Fully Computer-Verified Proof of Fermat's Last Theorem

A puzzle scrawled in the margin of a 17th century arithmetic textbook defied the world's best mathematicians for 358 years, and the proof that finally arrived in 1995 has now been rewritten as code that a computer can check on its own. Anthropic said on September 4 that Claude, its AI system, produced the first fully computer-checked formal proof of Fermat's Last Theorem, writing the argument in the Lean proof assistant over 11 days of largely autonomous work. The formalization translates Andrew Wiles's landmark 1995 proof into roughly 13 million lines of machine-verifiable code.

Fermat's Last Theorem is the deceptively simple statement that for any integer n greater than 2, no three positive integers a, b and c satisfy a^n + b^n = c^n. Fermat himself claimed around 1637 to have a marvelous proof that the margin of his book was too narrow to contain, and mathematicians now widely believe that proof, if it ever existed in rigorous form, was wrong, because the techniques needed to prove the theorem took another three and a half centuries to develop. Formalizing Wiles's argument means translating that human narrative into a syntax literal enough for software to inspect every inference. A step that a human reader accepts without thinking can expand into pages of explicit notation in a proof assistant, which is why this kind of work has traditionally consumed years of a specialist's time.

Anthropic said on September 4 that the result is verification rather than discovery, and the announcement carried an uncomfortable comparison for human formalizers. Kevin Buzzard, a mathematician at Imperial College London, launched a community project in 2024 to translate Wiles's proof into Lean, and that effort is funded through 2029 and still unfinished. Claude's run, launched overnight on August 7 and completed on the night of August 17, proved more than 30,000 supporting theorems along the way and consumed roughly 6 billion output tokens. Anthropic said on September 4 that the model behind the run was an internal research version roughly comparable to Claude Fable 5.1, the version the company later released to the public.

Key Facts

Anthropic released the complete proof on September 4 at github.com/anthropics/fermats-last-theorem under the permissive Apache 2.0 license, leaving the repository open for independent inspection. The company said the formalization relies only on Lean's three standard axioms and contains no unproved placeholders, meaning nothing is assumed without a machine check. The proof does not reproduce Wiles's original structure verbatim; Anthropic said on September 4 that it follows a simplified version of his strategy developed by Darmon, Diamond and Taylor.

The numbers give a sense of the scale. Anthropic reported on September 4 that Claude wrote 13 million lines of Lean code, more than five times the size of Mathlib, the principal community library of formalized mathematics on which the proof builds. The run generated computer-verifiable proofs of 30,300 theorems, of which 29,500 appear in the final argument. Decrypt reported on September 5 that the finished artifact is the longest math proof ever built and that, measured against an 80,000 word novel, it is the equivalent of roughly 160 novels of pure logical argument.

Human reviewers who have seen the proof are treating it seriously. Buzzard reviewed Claude's work and endorsed it, stating that it proves Fermat's Last Theorem with no assumptions other than the axioms of mathematics. Anthropic said on September 4 that Buzzard described the artifact as multi-layered and argued that AI autoformalization outputs are now robust enough for other researchers to build on. TechTimes reported on September 5 that the result does not represent new mathematics, since the underlying argument remains Wiles's, and that the distinction between verification and discovery may carry more weight for the field's future than a fresh theorem would.

Analysis

The bigger picture here is that proof assistants have crossed from a specialist niche into a tool that can absorb a body of work that mathematicians expected to take years. What changed is not a sudden leap in what a single model knows about number theory, but the scaffolding around the model. Anthropic said on September 4 that the first attempt failed because dozens of agents lost track of what they had already proved and stopped collaborating, and those false starts still account for roughly 7 percent of the lines in the final proof. The fix was an open platform called Prove2Me, built by a Columbia University group led by Anthropic researcher Tianyi Peng, which gave every agent a shared map of which theorems remained open.

What this really means is that the bottleneck in machine mathematics is coordination, not raw reasoning capacity. A single model cannot hold 13 million lines of context in mind at once, so the ability to record state outside any one agent is what made the marathon possible. TechTimes reported on September 5 that Prove2Me stored theorem statements separately from their proofs so that Lean only had to recompile files that actually changed, and that it kept plain English summaries of each result so agents could find relevant prior work by meaning rather than by scanning millions of lines. That division of labor, more than any individual lemma, is why an 11 day run could overtake a human project now in its third year.

Anthropic was explicit about what this achievement is not. On September 4 it noted that, unlike recent AI work that produced novel mathematics, the novelty here is verification, checking an existing proof the way a calculator checks an arithmetic sum. That distinction matters because mathematics is beginning to face a backlog of AI generated results that arrive faster than human referees can read them. Decrypt reported on September 5 that machine-checked proofs are deterministic and not prone to the human errors that can hide inside a hundred pages of dense argument, which is exactly the property the field needs as the volume of new mathematics grows.

Why It Matters

Fermat's Last Theorem is a cautionary tale about the limits of human review. Wiles announced a proof across three lectures in June 1993, only for reviewers to expose a critical gap during the verification effort, and he then spent close to a year repairing it with his former student Richard Taylor before publishing the corrected 129 page proof in May 1995. Anthropic said on September 4 that a single broken link can quietly falsify everything built on top of it and that formalization forces each link into the open, where software can examine it mechanically. The episode is a reminder that even the most careful readers can miss an error buried deep in a long argument.

The wider significance is that mathematics is turning into an AI production line, and formalization is the quality control layer. If models can render their own findings into machine-checkable form, the burden of evaluating new work, a process that Anthropic said on September 4 can consume months or even years of a mathematician's career, could shrink dramatically. The company said it hopes it will become easier, not harder, to trust the body of knowledge on which mathematics is built. The roughly 6 billion output tokens that Claude consumed are a reminder that this rigor is still costly, but the alternative, checking a proof of this difficulty by hand, can consume years of expert attention.

There is also a bracing benchmark in the raw numbers. Whatever one thinks of the surrounding hype, 13 million lines, 30,300 theorems and a proof more than five times the size of Mathlib reset expectations for what a frontier AI system can sustain over a long horizon. Anthropic emphasized that the argument rests only on Lean's three standard axioms, which narrows what a skeptic must distrust. And because the company published the repository under Apache 2.0, the artifact can be studied, reused and stress tested by the same community that maintains Mathlib, turning a corporate announcement into a public resource.

Next Up

What happens next depends largely on the mathematics community. Because the repository is public, other researchers can inspect Claude's lemmas, test whether they compile cleanly, and attempt to fold the most reusable parts into Mathlib. Buzzard, whose own formalization project remains funded through 2029, has signaled that he regards the artifact as a foundation rather than a terminal point, telling Anthropic, which published his remarks on September 4, that the proof is multi-layered and that AI formalization outputs are now robust enough to build upon. His endorsement gives the work a path into the mainstream formalization community.

The open questions are substantial. A single success on one of the hardest theorems in a generation does not guarantee that the recipe transfers cheaply to other grand proofs, and the appetite for roughly 6 billion output tokens raises genuine questions about cost. Anthropic said on September 4 that the winning configuration required the coordination platform, which suggests that reproducing the result depends as much on the scaffolding as on model strength. Even formalizing an existing proof at this scale demanded a novel division of labor, so applying the same machinery to conjectures with no human roadmap will require further advances in how agents share state and check one another's work.

Watch next for how formal methods reshape refereeing. If AI written mathematics is expected to clear a machine-checkable bar before it is trusted, the human referee's job shifts from hunting for buried errors toward judging whether a result is worth believing and building on. Anthropic framed the announcement on September 4 as a step toward a future in which all of mathematics can be readily checked. The speed of that transition now depends on whether tools like Prove2Me mature, whether Mathlib's maintainers integrate the new work, and whether university projects such as Buzzard's choose to stand on Claude's shoulders.