Pythagorean Triples in the Fibonacci Model Set

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.