On a Diophantine Equation with Jacobsthal and Fibonacci Numbers

Abstract

In the present paper, we identify all Jacobsthal numbers that may be expressed as a product of three Fibonacci numbers. More precisely, our main result shows that the only solution to the Diophantine equation \[ F_kF_lF_m=J_n \] for $2<k<l<m$ is \[ (k,l,m,n)=(5,7,8,12). \] The proof relies on techniques involving linear forms in logarithms.

12 pages. Final version. To appear in Hacettepe Journal of Mathematics and Statistics