Windmills of the minds: an algorithm for Fermat's Two Squares Theorem (2112.02556v2)

Abstract: The two squares theorem of Fermat is a gem in number theory, with a spectacular one-sentence "proof from the Book". Here is a formalisation of this proof, with an interpretation using windmill patterns. The theory behind involves involutions on a finite set, especially the parity of the number of fixed points in the involutions. Starting as an existence proof that is non-constructive, there is an ingenious way to turn it into a constructive one. This gives an algorithm to compute the two squares by iterating the two involutions alternatively from a known fixed point.

• The paper formalizes involutions by defining 'flip' and 'zagier' maps to convert Zagier’s proof into a constructive framework.
• It develops an iterative algorithm that alternates between these maps to locate fixed points, yielding the unique two-square representation for primes.
• Using the HOL4 theorem prover, the authors rigorously verify the algorithm’s correctness and demonstrate its efficiency through empirical tests.

Overview of "Windmills of the Minds: An Algorithm for Fermat's Two Squares Theorem"

This paper details the creation of an algorithm based on Zagier's proof for Fermat's Two Squares Theorem. It formalizes the proof using involutions and develops a constructive approach to verify Fermat's claim that a prime number $n$ of the form $4k + 1$ can be uniquely expressed as the sum of two squares. The paper's core innovation lies in its conversion of Zagier's proof into an algorithm using iterative theorem proving in the HOL4 system.

Key Contributions

1. Formalization of Involutions: The paper formalizes the concept of involutions on a set, allowing verification that the Zagier map is an involution. Two specific involutions, flip and zagier, are defined over the set of windmills (ordered triples) to establish the theorem constructively.
2. Algorithm Development: An algorithm is derived from the interplay between the two involutions. By initiating from a known Zagier fixed point, iterating between the flip and zagier maps locates a fixed point of the flip map, which provides the two squares.
3. Iterative Theorem Proving: The authors employed iterative theorem proving techniques to validate the algorithm. The period of the iterative cycle of involutions determines termination and correctness of the algorithm. The HOL4 theorem prover ensures the accuracy of these formalizations.
4. Empirical Evaluation: The paper demonstrates the practicality of the algorithm with experimental results. These experiments confirm its efficacy in calculating the two squares for large primes efficiently within the HOL4 environment.

Theoretical Insights and Implications

• Relation to Zagier's Proof: By deriving an algorithm through formal proof manipulation, the paper provides deeper insight into Zagier’s proof mechanics, showcasing the feasibility of translating a non-constructive proof into a practical computational method.
• Formal Verification: The work highlights the potential for using formal verification in mathematics to derive algorithms directly from proofs, ensuring both correctness and a clear reflection of the underlying theoretical principles.
• Potential for Broader Applications: The theoretical framework developed could extend beyond this theorem, offering a template for addressing other problems in number theory through iterative involution techniques and theorem proving.

Future Directions

The paper suggests further exploration into performance optimization by leveraging group action symmetries and applying similar methodologies to other theorems where formal proof might yield effective algorithms. There's also interest in combining this approach with existing cryptographic and computational number theory efforts to address both theoretical and applied problems more rigorously.

Through this work, the authors not only solve a historic problem using modern tools but also set a precedent for future computational approaches in mathematics, demonstrating the viability of merging classical theory with contemporary computational logic frameworks.