A complete formalization of Fermat's Last Theorem for regular primes in Lean (2410.01466v3)

Abstract: We formalize a complete proof of the regular case of Fermat's Last Theorem in the Lean4 theorem prover. Our formalization includes a proof of Kummer's lemma, that is the main obstruction to Fermat's Last Theorem for regular primes. Rather than following the modern proof of Kummer's lemma via class field theory, we prove it by using Hilbert's Theorems 90-94 in a way that is more amenable to formalization.

• The paper formalizes Fermat’s Last Theorem for regular primes using Lean, rigorously integrating algebraic number theory with formal proof methods.
• The work develops extensive number theory libraries, detailing cyclotomic fields, discriminants, and ramification to support formal verification.
• The application of Kummer’s Lemma and Hilbert’s theorems streamlines complex classical proofs, paving the way for future enhancements in formal verification.

Overview of the Formalization of Fermat's Last Theorem for Regular Primes in Lean

The paper "A complete formalization of Fermat's Last Theorem for regular primes in Lean" presents a significant contribution to the formalization of mathematical proofs using the Lean theorem prover, specifically focusing on Fermat's Last Theorem (FLT) for regular primes. This work integrates computer science and algebraic number theory, offering insights into formal proof environments.

Key Contributions

The authors formalize a complete proof for Fermat's Last Theorem in the case of regular primes using the Lean theorem prover. Regular primes are those that do not divide the class number of the cyclotomic extension $\mathbb{Q}(\zeta_p)$, where $\zeta_p$ is a primitive $p$-th root of unity. The paper targets the formalization of mathematical concepts like cyclotomic fields, units in these fields, and the intricacies of number theory—an area that traditionally relies heavily on informal reasoning and intuition.

Formalization Highlights

1. Cyclotomic Fields and Regular Primes:
   • The formalization starts with cyclotomic fields and extensions, providing explicit examples of number fields within Lean's framework.
   • Definition of regular primes is incorporated as it relates to the class number, simplifying the number-theoretical considerations that ensue.
2. Case 1 and Development of Number Theory Libraries:
   • The paper outlines the proof of the first case of FLT for regular primes, which involves developing extensive number theory libraries in Lean.
   • This includes formalizing concepts such as discriminants, ramification, and cyclotomic fields, significantly enriching Lean's mathematical library.
3. Case 2 and Induction via Kummer's Lemma:
   • The reduction to Kummer's Lemma is pivotal for handling the second case of FLT. The authors detail an elementary proof that avoids full reliance on class field theory, making it more suitable for formalization.
   • The key difficulty lies in adapting these classical proofs to fit within the constraints of formal proof systems while maintaining readability and correctness.
4. Hilbert's Theorems and Ramification:
   • The paper uses Hilbert's Theorems 90-94 to address critical aspects of the proof, circumventing more complex theories that are not yet formalizable within Lean.
   • The formalization of unramified extensions and related group cohomology provides a foundation for this approach, showcasing the potential for growth in formal verification of algebraic number theory.

Numerical and Theoretical Insights

The authors present strong evidence through their formalization that the tools available in Lean can verify profound mathematical theorems. The formal proof not only confirms the validity but also finds underlying mathematical structures inherently suitable for computational logic. While the focus remains on formalizing math for Lean, this work underscores the capability of modern proof assistants to engage with complex mathematical problems once considered beyond their scope.

Implications and Future Directions

The implications are twofold: enriching the mathematical capabilities of Lean and presenting a pathway for formalizing results that drive mathematical intuition and development. Furthermore, this paper bridges the gap between classical number theory and modern computational methods, suggesting future integration of class field theory, potentially extending formal verification to even more complex theorems.

Looking forward, the authors indicate potential developments in formalizing the $p$-adic class number formula, which would entail computations with Bernoulli numbers and the $p$-adic $L$-functions—projects that promise to stretch the boundaries of what can be achieved within current computational frameworks.

Overall, this paper demonstrates, with rigor and precision, how classical mathematics can interface with computational tools to further both fields. This work stands as a testament to the ongoing evolution of formal methods in capturing and verifying mathematical truth.