Anthropic’s Claude model generated over a million lines of Lean code to formalize Wiles’s proof of Fermat’s Last Theorem.
ANTHROPIC'S AI FORMALIZES FERMAT'S LAST THEOREM, SETTING NEW BAR FOR MACHINE‑PROVED MATHEMATICS
*Anthropic’s Claude model completed a Lean formalization of Andrew Wiles’s proof in 48 hours. The feat reshapes AI’s role in high‑assurance mathematics and raises alarms about the future of human mathematicians.*
Anthropic’s AI team announced on September 4, 2026 that they have fully formalized Andrew Wiles’s proof of Fermat’s Last Theorem in the Lean theorem prover. The result, posted on the company’s research blog and mirrored on GitHub, marks the first time an artificial‑intelligence system has completed a machine‑checked proof of a millennium‑era theorem. Claude 3.0 consumed roughly 2.4 PFLOPs of compute over a continuous 48‑hour run to translate Wiles’s 1995 proof into 1.3 million lines of Lean code.
The breakthrough collapses the historic gap between human‑crafted proofs and formal verification. Anthropic’s engineers wrote 12,000 auxiliary lemmas and a custom tactic library to bridge gaps in existing math libraries. The effort required a dedicated cluster of 64 GPU‑accelerated nodes, each running at full capacity.
Skeptics warn the milestone fuels a narrative that AI can replace mathematicians. Proponents argue the technology unlocks error‑free verification for cryptography, quantum algorithms, and safety‑critical software. The stakes are clear: AI‑driven formal methods could become a new standard for high‑assurance engineering.
The Technical Breakthrough
Claude 3.0 leveraged a fine‑tuned transformer architecture to parse Wiles’s 1995 paper, extract logical dependencies, and generate Lean statements. The model iteratively proposed lemmas, which human overseers vetted for semantic correctness before committing to the repository. Over 1.3 million lines of code, the system reconstructed the modular elliptic curve argument, the Taniyama‑Shimura link, and the final descent step. A custom tactic library, written in Lean’s meta‑programming language, reduced proof search time by 70 % compared with baseline tactics. The result passed Lean’s kernel verifier without any human‑identified gaps, a milestone previously achieved only for elementary theorems.
Compute Costs and Scaling
Anthropic logged 2.4 PFLOPs of floating‑point work, equivalent to 1.2 MWh of electricity, for the 48‑hour run. The expense translates to roughly $15,000 in cloud credits on Anthropic’s internal pricing. Scaling the pipeline to more complex conjectures—such as the Langlands program—could require an order of magnitude more compute. The team estimates a 30 % cost reduction per theorem after refining the tactic library and adding a reinforcement‑learning loop that prunes unproductive search branches. Even with these efficiencies, the price tag remains prohibitive for most academic labs, cementing a competitive advantage for well‑funded AI firms.
Industry Reaction and Competitive Landscape
Microsoft’s DeepMind released a partial formalization of the ABC conjecture in March 2026, but stopped short of a full proof due to tactic limitations. Google AI’s AlphaTensor team announced a separate effort to formalize matrix multiplication lower bounds, citing Anthropic’s success as a catalyst. Venture capitalists have already earmarked $200 million for “AI‑first theorem proving” startups. At the same time, the International Mathematical Union issued a statement urging caution, emphasizing that formal verification should augment—not replace—human insight. The race is intensifying, with each lab racing to claim the next “millennium theorem” as a benchmark.
Implications for Research and Regulation
A fully verified proof of FLT removes any lingering doubt about hidden errors, setting a precedent for critical domains like cryptographic protocol design. Regulators in the EU and US are drafting guidelines that could mandate formal verification for software governing autonomous vehicles and medical devices. If AI can deliver such verification at scale, compliance costs could drop dramatically, but the monopoly over high‑end compute may concentrate power in a few tech giants. Academics warn that reliance on black‑box models may erode the training pipeline for future mathematicians, potentially creating a talent vacuum.
Anthropic’s formalization of Fermat’s Last Theorem proves that AI can now shoulder the most intricate logical burdens once reserved for human genius. The immediate payoff is undeniable: error‑free verification for high‑stakes engineering. The longer‑term risk is a reshaped research ecosystem where access to petaflop‑scale clusters dictates who can claim mathematical breakthroughs. As governments draft AI‑safety standards, the industry must decide whether to democratize the tooling or let a handful of corporations monopolize the future of proof.
Sources: Anthropic Research Blog (https://www.anthropic.com/research/formalizing-fermat’s-last-theorem), Hacker News discussion thread, Xena Project blog post (https://xenaproject.wordpress.com/2026/09/04/flt-anthropic-has-beaten-me-to-it)