On the diophantaine equations $J_N+J_M=F_A$ & $F_N+F_M=J_A$

Abstract

Let ${\left\{F_k\right\}}_{k\ge 0}$ be the Fibonacci sequence defined by $F_k=F_{F-1}+F_{k-2}$ for all $n\ge 2$ with initials $F_0=0F_1=1$. Let ${\left\{J_n\right\}}_{n\ge 0}$ be the Jacobsthal sequence defined by $J_n=2J_{n-1}+J_{n-2}$ for all $n\ge 2$ with initials $J_0=0$, $J_1=1$. In this paper we find all the solutions of the two Diophantine equations $J_n+J_m=F_a$ ,$F_n+F_m=J_a$ in the non-negative integer variables (n,m,a),i.e we determine all Fibonacci numbers which are sum of two Jacobsthal numbers, and also determine all Jacobsthal numbers which are sum of two Fibonacci numbers.