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

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.