Formalizing Fermat's Last Theorem

On June 24, 2023, Anthropic researchers announced the completion of the first fully computer-checked proof of Fermat's Last Theorem (FLT). Claude, one of Anthropic's AI systems, autonomously created the proof in 11 days, writing an astounding 13 million lines of code in the Lean programming language. Claude proved 29,500 intermediate theorems along the way to create the final proof, which is 5 times larger than Mathlib, a principal library of mathematical proofs.

Fermat's Last Theorem, proposed by Pierre de Fermat in 1637, states that no positive integers a, b, and c can satisfy the equation a^n + b^n = c^n for any integer value of n greater than 2. The theorem remained unproven for over 350 years, despite numerous attempts and a 100,000 German gold mark prize offered in 1908 for a correct proof.

In 1995, Sir Andrew Wiles finally proved FLT, but his proof required 129 pages and was left to be verified for months by numerous mathematicians. The proof heavily relied on modern mathematical techniques that were not available in Fermat's time. In June 1993, Wiles's proof was exposed to a critical gap during verification, leading to a year of intense effort to fix it, which finally succeeded in May 1995 with the assistance of Richard Taylor.

The automated proof created by Claude represents a significant step towards the future where all of mathematics can be readily checked using AI. This achievement has the potential to lighten the burden of evaluating new results, as the process of formalizing a proof can take years. However, unlike recent AI-driven work on the Riemann hypothesis that produced novel mathematics, Claude's proof is a verification of an existing theorem.