On a cubic family of Thue equations involving Fibonacci numbers and powers of two

Abstract

In this paper we completely solve the family of parametrised Thue equations (Figure presented.) where Fn is the n-th Fibonacci number. In particular, for any integer n ≥ 3 the Thue equation has only the trivial solutions (±1; 0); (0;∓1);∓(Fn; 1);∓(2 n; 1).