On a cubic Family of Thue Equations involving Fibonacci Numbers and Powers of Two

Ingrid Vukusic

paper

On a cubic Family of Thue Equations involving Fibonacci Numbers and Powers of Two

arXiv:2107.03097

Abstract

In this paper we completely solve the family of parametrised Thue equations \[ X(X-F_n Y)(X-2^n Y)-Y^3=\pm 1, \] where $F_{n}$ is the $n$ -th Fibonacci number. In particular, for any integer $n \geq 3$ the Thue equation has only the trivial solutions $(\pm 1, 0), (0, \mp 1), \mp (F_{n}, 1), \mp (2^{n}, 1)$ .