On split families of Thue equations with linear recurrence sequences as factors

TL;DR

This paper investigates a family of Thue equations involving linear recurrence sequences as factors, extending known results from polynomial parameters to these more complex sequences, and confirms solutions are explicitly describable.

Contribution

It generalizes the explicit solution results of Thue equations from polynomial parameters to certain linear recurrence sequences.

Findings

Solutions are explicitly describable for the considered family.

The result extends known polynomial cases to linear recurrence sequences.

Confirmed the existence of explicit solutions for the new family.

Abstract

We consider a parametrised family of Thue equations, \[ (x-G_1(n)\, y) \cdots (x-G_d(n)\, y) - y^d = \pm 1, \] which was first considered by Thomas to have an explicit set of solutions for parameters $n$ larger than some effectively computable constant. In the case where the parameter functions are polynomials belonging to an explicitly described family, this is known to be true. We consider other parameter functions, namely linear recurrence sequences, for which it is not obvious that a similar result holds, and confirm that it does for an explicitly described family of linear recurrence sequences.