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
X(X - F_nY )(X - 2ⁿY ) - Y³ = ± 1,
where F_n 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), ∓(F_n,1), ∓(2ⁿ,1).