Diophantine equations related to quasicrystals: a note

TL;DR

This paper solves three Diophantine equations within the ring of integers of the algebraic number field Q[√5], addressing their connection to minimum distance problems in fivefold symmetric quasicrystals.

Contribution

It provides the general solutions to these equations, linking algebraic number theory with quasicrystal structure analysis.

Findings

Explicit solutions to three Diophantine equations in Q[√5]

Insights into minimum distance problems in fivefold symmetric quasicrystals

Bridges between algebraic number theory and quasicrystal geometry

Abstract

We give the general solution of three Diophantine equations in the ring of integer of the algebraic number field ${\bf Q}[{\sqr 5}]$. These equations are related to the problem of determination of the minimum distance in quasicrystals with fivefold symmetry.