Norm Form Equations and Linear Divisibility Sequences

TL;DR

This paper explores how solutions to norm form equations over quartic fields can be transformed so that specific solution sequences exhibit linear divisibility properties, linking algebraic number theory with recurrence sequences.

Contribution

It demonstrates that for certain quartic norm forms, one can find equivalent forms making a fixed coordinate sequence a linear divisibility sequence.

Findings

Solutions form tuples of linear recurrence sequences

Existence of equivalent forms with divisibility properties

Application to quartic fields

Abstract

Finding integer solutions to norm form equations is a classical Diophantine problem. Using the units of the associated coefficient ring, we can produce sequences of solutions to these equations. It is known that these solutions can be written as tuples of linear recurrence sequences. We show that for certain families of norm forms defined over quartic fields, there exist integrally equivalent forms making any one fixed coordinate sequence a linear divisibility sequence.