The Digital Verification of Fermat's Last Theorem

On September 4, 2026, Anthropic announced a landmark achievement in the intersection of artificial intelligence and mathematics: the first fully computer-verified proof of Fermat’s Last Theorem (FLT).

Working with a high degree of autonomy over an 11-day period, the AI model Claude authored the proof using the Lean programming language. This milestone represents a shift in how complex mathematical research can be validated.

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📜 The Legacy of the Conjecture

The story begins around 1637, when Pierre de Fermat wrote a famous note in the margin of Diophantus’s Arithmetica. He claimed that for any integer $n > 2$, there are no positive integers $a, b, \text{ and } c$ that satisfy the equation:

$\text{a}^n + \text{b}^n = \text{c}^n$

The Long Road to Proof

For centuries, this claim remained an elusive puzzle. The difficulty of FLT is illustrated by the following timeline:

The quest was fraught with failure. In the first year after the 1908 prize announcement, 621 incorrect attempts were submitted. Even Sir Andrew Wiles, who finally cracked the code in 1995 with a 129-page proof, faced a grueling verification process. After his 1993 lectures, a reviewer found a critical flaw, forcing Wiles and Richard Taylor to spend another year refining the logic before the final publication.

"I have discovered a truly marvelous proof of this, which this margin is too narrow to contain." — Pierre de Fermat

Because the eventual proof relied on modern techniques unknown in the 17th century, most mathematicians now believe Fermat's original "marvelous proof" was actually incorrect.

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💻 The Challenge of Formalization

While Wiles's proof was accepted by humans, "formalizing" it means translating it into a language that a computer can check with absolute certainty. This is where tools like the Lean proof assistant come in.

Human Proofs vs. Formal Proofs

To achieve this, the community—led by Kevin Buzzard of Imperial College London since 2024—began a massive effort to encode the necessary mathematical framework. The blueprint for this project alone spanned 86 pages.

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🤖 Claude's Contribution

Tianyi Peng, a researcher at Anthropic and Columbia University, tested Claude's ability to handle this complexity. The results were staggering.

The Technical Scale

Claude didn't just summarize the proof; it constructed it from the ground up.

• Duration: 11 days.
• Volume: 13 million lines of Lean code.
• Complexity: 30,300 theorems generated (29,500 used in the final chain).
• Scale: The resulting proof is over 5x larger than Mathlib, the primary community library for Lean.

The Process

Claude utilized a multi-agent architecture to tackle the problem:

• Define fundamental mathematical concepts.
• Establish intermediate lemmas.
• Chain lemmas to prove increasingly complex statements.
• Finalize the end-to-end proof of FLT.

-- Conceptual example of a Lean statement
theorem fermat_last_theorem (n : ℕ) (a b c : ℕ) 
  (hn : n > 2) (habc : a ≠ 0 ∧ b ≠ 0 ∧ c ≠ 0) : 
  ¬ (a^n + b^n = c^n) := 
begin
  -- Claude's 13 million lines of logic would go here
  sorry, 
end

![Image Placeholder: A visualization of the multi-layered structure of the AI-generated proof]

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🚀 Implications for Mathematics

This achievement is not about discovering new math (unlike previous AI work on the Riemann hypothesis), but about verification.

Kevin Buzzard noted the significance of this work:

"This extraordinary autoformalization achievement... proves Fermat’s Last Theorem with no assumptions other than the axioms of mathematics. Along the way we see autoformalization of algebra, harmonic analysis, geometry and number theory..."

Why This Matters

1. Reduced Burden: Verifying a complex new result currently takes years of human effort. Autoformalization could make this process nearly instantaneous.
2. Robustness: The proof is "multi-layered," meaning AI-generated mathematical artifacts are now stable enough to serve as foundations for further research.
3. Trust: It ensures that the body of mathematical knowledge is built on a foundation that is actually correct, rather than just "generally accepted."

Claude's approach followed a streamlined version of the proof provided by Darmon, Diamond, and Taylor, requiring only minimal high-level guidance from human researchers.