Pythagorean Triples in the Fibonacci Model Set

TL;DR

This paper characterizes Pythagorean triples within the Fibonacci model set and explores solutions to Fermat's Last Theorem in this context, including a counterexample for the third degree.

Contribution

It provides a complete description of Pythagorean triples in the ring Z[τ] and investigates Fermat's equation solutions in the Fibonacci model set.

Findings

All Pythagorean triples in Z[τ] are described.

A counterexample to Fermat's Last Theorem for n=3 is presented.

Examples illustrate the solutions within the Fibonacci model set.

Abstract

In this paper we give a description of all Pythagorean triples in the ring ${{\mathbb Z}}[\tau]$. We also consider triples in the Fibonacci model set which satisfy the Diophantine equations arising from Fermat's Last Theorem. Examples are provided, including a counterexample to Fermat's Last Theorem for the third degree in the Fibonacci model set.