Anthropic says its Claude model formalised Andrew Wiles’s 1995 proof of Fermat’s Last Theorem into 13 million lines of machine-checkable Lean code, a translation task mathematicians expected to take years, completing it in 11 days and prompting number theorists to call the result the largest formal proof ever constructed. NSH is running this under its continued-press-attention provision: the breakthrough was announced on 4 September, over a week before this cycle, but is included because expert commentary and analysis — including a Nature feature published four days ago — are still actively shaping how the finding is being read.
On 4 September 2026, Anthropic announced that its Claude model had produced the first complete, end-to-end formalisation of Fermat’s Last Theorem — meaning Andrew Wiles’s celebrated 1995 proof of the centuries-old conjecture has now been converted into Lean, a formal language whose every logical step can be automatically checked by computer, with no gaps a human reviewer has to trust on faith. The effort, led by Anthropic researcher Tianyi Peng’s formalisation group (which is affiliated with Columbia University), took 11 days of largely autonomous work by Claude, working across several dozen parallel agents that together generated roughly 6 billion tokens. Human formalisation experts had previously estimated the task — not proving the theorem, which Wiles had already done, but converting the proof into fully machine-checkable form — would take years of dedicated specialist labour.
It is important to be precise about what was and was not achieved. Fermat’s Last Theorem — that no three positive integers a, b, c can satisfy a^n + b^n = c^n for any integer n greater than 2 — was proven by Andrew Wiles in 1995, closing a problem first posed by Pierre de Fermat in 1637. Claude did not discover new mathematics or improve on Wiles’s argument. What it did was translate that existing, already-accepted proof into Lean 4 code that a computer proof assistant can verify mechanically, line by line, rather than relying on the years of painstaking human peer review that verifying Wiles’s original proof required. Along the way, Claude produced and formally verified 29,500 intermediate theorems — many covering areas of mathematics that had never previously been formalised at all — resulting in a codebase Anthropic describes as the largest Lean proof ever constructed, at over 13 million lines.
Why formalisation is hard, and why the speed matters
Formalising a major proof is notoriously slow, exacting work: every implicit step a human mathematician takes for granted has to be spelled out in a form a computer can check, and gaps or ambiguities that a human reviewer would silently resolve become hard failures in a formal system. Kevin Buzzard, the Imperial College London mathematician who reviewed Claude’s output and has separately worked for years on formalising number theory in Lean, called it “an extraordinary autoformalization achievement,” noting the proof relies on no assumptions beyond the standard axioms of mathematics. Alex Kontorovich, a number theorist at Rutgers University, said watching a machine convert decades of human mathematical work into an ironclad, computer-checked proof “completely blew my mind.” According to reporting on the underlying technical process, the first formalisation attempt by Claude’s agents actually failed: early multi-agent runs collapsed because individual agents accumulated too much local context, lost track of results already proved elsewhere in the effort, and duplicated work across the dependency graph. Progress became possible only once Prove2Me — an open-source formalisation tool built at Columbia — was incorporated mid-run, alongside an orchestration layer to manage a shared dependency graph across agents, suggesting the achievement rests as much on systems architecture as on any single model capability.
Why it matters
The direct beneficiaries of this specific formalisation are narrow — Fermat’s Last Theorem itself has few practical applications, and its 1995 proof was already accepted by the mathematical community. The broader significance, which is why the story has continued generating expert commentary a full week after the initial announcement, is what it demonstrates about formal verification at scale. Mathematical peer review currently depends on human experts reading and re-deriving a proof’s logic, a process that took months even for specialists checking Wiles’s original 1995 proof, and one that history shows is imperfect — flawed proofs have gone unchallenged for years before being caught. If an AI system can now formalise proofs of this scale and complexity in days rather than years, it raises a genuine prospect, one Anthropic has stated explicitly as an ambition, that future mathematical results could be submitted for review paired with a machine-verified formal version from the outset, changing how mathematical claims are checked and trusted industry-wide. Buzzard’s own reaction captures the stakes: if automatic formalisation of a proof this large is achievable now, it suggests a genuine path toward formalising a much larger share of the existing mathematical literature — work that, until now, has depended on a small, specialised community of human formalisers making slow, manual progress. The appropriately sceptical caveat is that this is one result on one already-proven theorem; whether the same speed and reliability generalise to formalising genuinely new, not-yet-verified mathematical claims — where there is no existing accepted proof to check the formalisation against — remains to be demonstrated.
Veer Reddy Pulagam
Key facts
- Anthropic’s Claude produced the first complete, machine-checked Lean 4 formalisation of Fermat’s Last Theorem, announced 4 September 2026
- Task: converting Andrew Wiles’s already-accepted 1995 proof into fully machine-verifiable form, not discovering new mathematics
- Completed in 11 days of largely autonomous multi-agent work; produced 13+ million lines of Lean code and 29,500 verified intermediate theorems — the largest formal Lean proof to date
- Reviewed and praised by Imperial College London mathematician Kevin Buzzard; first formalisation attempt failed before Columbia’s Prove2Me tool was incorporated mid-run
- Continued expert discussion as of 8 September 2026 (Nature feature), a week after the original announcement
