On families of cubic split Thue equations parametrised by linear recurrence sequences

TL;DR

This paper proves that for large enough parameters, a family of cubic split Thue equations parametrized by linear recurrence sequences has only trivial solutions, extending understanding of solutions to these equations under certain conditions.

Contribution

It establishes the finiteness of solutions for a family of cubic split Thue equations parametrized by linear recurrence sequences, under specific dominant root and technical conditions.

Findings

Only trivial solutions exist for sufficiently large parameters.

The solutions are explicitly characterized as .

The result applies to sequences meeting dominant root and technical conditions.

Abstract

Let $(A_n)_{n\in \mathbb{N}}, (B_n)_{n\in \mathbb{N}} \in \mathbb{Z}^{\mathbb{N}}$ be two linear-recurrent sequences that meet a dominant root condition and a few more technical requirements. We show that the split family of Thue equations \[ |X(X-A_n Y)(X-B_n Y) - Y^3| = 1 \] has but the trivial solutions $\pm\{ (0,1), (1,0), (A_n,1), (B_n,1) \}$, if the parameter $n$ is larger than some effectively computable constant.