Fermat's Last Theorem (FLT) stands as one of mathematics' most famous challenges, taking over 350 years to solve. Now, an ambitious project led by Professor Kevin Buzzard at Imperial College London aims to formalize this monumental proof in the Lean theorem prover, marking a significant milestone in the intersection of mathematics and formal verification.
The Challenge of Formalizing FLT
While the statement of Fermat's Last Theorem is remarkably simple: if x, y, z, and n are positive integers with n ≥ 3, then x^n + y^n ≠ z^n. However, the proof is notoriously complex. Andrew Wiles' breakthrough proof in the 1990s, completed with Richard Taylor, draws on numerous areas of mathematics, such as:
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Algebraic and analytic number theory
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Algebraic and differential geometry
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Commutative algebra
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Harmonic analysis
The FLT formalization project isn't tackling the original Wiles/Taylor-Wiles proof but a "21st century" version that incorporates subsequent developments by Khare-Wintenberger, Kisin, and others. At its core remains the revolutionary "R = T" concept—that a deformation ring is isomorphic to a Hecke algebra, which was the key insight in Wiles' approach.
Project Organization and Research Focus
The project is organized using Patrick Massot's blueprint software, creating a comprehensive map of the theorem and LaTeX documentation of the proof. This structure makes the project accessible to mathematicians at various levels of expertise with Lean.
Rather than starting with the basics, the project focuses on formalizing advanced mathematical objects and techniques used in contemporary research. This approach ensures that:
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New definitions and theories are contributed to Lean's Mathlib library.
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Graduate students in number theory can engage with formalization through accessible projects.
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Formal definitions of objects studied in modern number theory become available in theorem provers.
Why This Project Matters
For research mathematicians and organizations interested in formal verification, the FLT project demonstrates several key benefits of Lean:
For Research Mathematicians
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New Research Tools: The project is digitizing numerous mathematical objects and techniques used in modern research, making them available for new applications.
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Collaboration Platform: The modular approach enables mathematicians to collaborate on a massive formalization project without requiring expertise in the entire proof.
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Educational Resource: The growing repository of formalized mathematics provides an error-free reference for students and researchers learning advanced number theory.
For Organizations
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Scalability Demonstration: The project shows how Lean can handle extremely complex mathematical assertions that span thousands of pages of informal mathematics.
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Training Data: The formalized proof will generate high-quality training data for AI systems that aim to assist with mathematical reasoning.
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Verification Benchmark: As the last remaining item in Freek Wiedijk's list of 100 challenge problems for computer formalization, FLT represents a significant benchmark for formal verification technology.
Getting Involved
The formalization of Fermat's Last Theorem is a massive challenge in formal mathematics that advances the frontier of what can be formally verified while creating a valuable resource for mathematicians and computer scientists.
The FLT formalization project welcomes contributors at all levels of expertise. Follow the project's progress on GitHub or explore the project website to learn more about this important formalization effort.