Formalizing Fermat's Last Theorem

Fermat's Last Theorem (FLT), an elusive mathematical conjecture, claimed that no positive integers a, b, and c can satisfy the equation a^n + b^n = c^n for any integer n greater than 2. Proposed by Pierre de Fermat in 1637, the theorem remained unproven for over three centuries, attracting the efforts of countless mathematicians.

In 1995, Sir Andrew Wiles finally proved FLT, but his proof spanned 129 pages and required years of verification. To streamline the process of verifying complex mathematical proofs, researchers began "formalizing" them, encoding the reasoning in a form that computers could automatically check. Dutch computer scientist Jan Bergstra first proposed this approach in 1995.

In recent years, Kevin Buzzard at Imperial College London and others have made significant progress in formalizing FLT using the Lean proof assistant—a formal verification tool. Recently, Tianyi Peng from Anthropic tested Claude, Anthropic's AI system, on the task of formalizing FLT. Remarkably, Claude achieved this in just 11 days, autonomously writing 13 million lines of Lean code and proving 29,500 intermediate theorems.

Upon completion, Kevin Buzzard, a renowned expert in the field, praised Claude's work, stating that the proof verifies FLT's truth using only the axioms of mathematics. This groundbreaking achievement signifies a major step towards the verification of all mathematical knowledge, potentially reducing the time and effort required to evaluate new results.

Unlike recent AI-driven work on the Riemann hypothesis, which produced novel mathematics, Claude's contribution is purely verification-based. The novelty lies in the ability to check a mathematical proof with the same rigor as one would verify a mathematical computation using a calculator. Fermat's Last Theorem serves as an illustrative example of the complexity and depth required in mathematical proofs.

Fermat's assertion, "I have discovered a truly marvelous proof...," remained unverified for over 350 years, until Wiles's 1995 proof. In 1908, a prize of 100,000 German gold marks was offered for a correct proof, leading to 621 incorrect attempts in just the first year. Wiles's initial proof was later found to contain a critical gap, which he and his former student Richard Taylor spent a year correcting.

Ultimately, the first correct proof was published in 1995, relying on advanced mathematical techniques far beyond Fermat's era. Today, mathematicians believe Fermat's original "marvelous proof" to be incorrect. Automating the verification process through proof assistants like Lean offers a means to ensure the correctness of complex proofs with absolute certainty.

The challenge for humans lies in translating proofs into a form that computers can understand, often requiring extensive manual effort. For FLT, the formalization process was anticipated to span years due to the need to build upon existing formalized mathematical concepts. Claude's proof, consisting of 13 million lines of Lean code, surpasses the size of the primary community library, Mathlib, by five times.

Collaborating with dozens of Claude agents, the project defined concepts, proved intermediate theorems, and constructed an elaborate multi-layered proof. Although human input was limited to occasional high-level instructions, Claude's autonomous work showcases the potential of AI in enhancing and accelerating mathematical research.