On the Diophantine Equations $J_n +J_m =L_k$ and $L_n +L_m =J_k$

Abstract

This paper finds all Lucas numbers which are the sum of two Jacobsthal numbers. It also finds all Jacobsthal numbers which are the sum of two Lucas numbers. In general, we find all non-negative integer solutions $(n, m, k)$ of the two Diophantine equations $L_n +L_m =J_k$ and $J_n +J_m =L_K,$ where $\left\lbrace L_{k}\right\rbrace_{k\geq0}$ and $\left\lbrace J_{n}\right\rbrace_{n\geq0}$ are the sequences of Lucas and Jacobsthal numbers, respectively. Our primary results are supported by an adaption of the Baker's theorem for linear forms in logarithms and Dujella and Peth\H{o}'s reduction method.