On a family of cubic Thue Equations involving Fibonacci and Lucas numbers

Abstract

Let $F_n$ denote the $n$-th Fibonacci number and $L_n$ the $n$-th Lucas number. We completely solve the family of cubic Thue equations $$(X-F_{n}Y)(X-L_{n}Y)X-Y^3=\pm 1$$ and show that there are no non-trivial solutions for $n\ne 1,3$.