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

TL;DR

This paper completely solves a family of parametrized Thue equations involving Fibonacci numbers and powers of two, identifying all solutions for all integers n ≥ 3.

Contribution

It provides a complete solution to a specific family of Thue equations involving Fibonacci numbers and powers of two, which was previously unresolved.

Findings

Only trivial solutions exist for n ≥ 3

Explicit solutions are identified for all n ≥ 3

The family of equations is fully characterized

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 $(\pm 1,0), (0,\mp 1), \mp(F_n,1), \mp(2^n,1)$.