Windmills of the minds: an algorithm for Fermat's Two Squares Theorem

12/05/2021
by   Hing Lun Chan, et al.
0

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.

READ FULL TEXT

page 1

page 2

page 3

page 4

research
12/18/2021

A Machine-Checked Direct Proof of the Steiner-Lehmus Theorem

A direct proof of the Steiner-Lehmus theorem has eluded geometers for ov...
research
03/11/2019

How far away must forced letters be so that squares are still avoidable?

We describe a new non-constructive technique to show that squares are av...
research
06/17/2020

Computer-assisted proofs for Lyapunov stability via Sums of Squares certificates and Constructive Analysis

We provide a computer-assisted approach to ensure that a given continuou...
research
06/11/2022

Householder Meets Student

The Householder algorithm for the QR factorization of a tall thin n x p ...
research
10/13/2021

Where are the logs?

The commonly quoted error rates for QMC integration with an infinite low...
research
10/18/2019

Minimal automaton for multiplying and translating the Thue-Morse set

The Thue-Morse set T is the set of those non-negative integers whose bin...
research
04/26/2021

The Relative Consistency of the Axiom of Choice Mechanized Using Isabelle/ZF

The proof of the relative consistency of the axiom of choice has been me...

Please sign up or login with your details

Forgot password? Click here to reset