  Let $\left\lbrace F_{k}\right\rbrace_{k\geq0}$ be the Fibonacci sequence
defined by $F_{k}=F_{F-1}+F_{k-2}$ for all $ n\geq2$ with initials $F_{0}=0\;
F_{1}=1$. Let $\left\lbrace J_{n}\right\rbrace_{n\geq0}$ be the Jacobsthal
sequence defined by $J_n=2J_{n-1}+J_{n-2}$ for all $ n\geq2$ 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.

Number Theory