Fibonacci numbers which are products of two Jacobsthal numbers

Abstract

In this paper, we find all Fibonacci numbers which are products of two Jacobsthal numbers. Also we find all Jacobsthal numbers which are products of two Fibonacci numbers. More generally, taking $k,m,n$ as positive integers, it is proved that $F_{k}=J_{m}J_{n}$ implies that \begin{align*} (k,m,n) = &(1,1,1),(2,1,1),(1,1,2),(2,1,2),\\ & (1,2,2),(2,2,2),(4,1,3),(4,2,3),\\ & (5,1,4),(5,2,4),(10,4,5),(8,1,6),(8,2,6) \end{align*} and $J_{k}=F_{m}F_{n}$ implies that \begin{align*} (k,m,n) =&(1,1,1),(2,1,1),(1,2,1),(2,2,1),\\ & (1,2,2),(2,2,2),(3,4,1),(3,4,2),\\ & (4,5,1),(4,5,2),(6,8,1),(6,8,2). \end{align*}