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

TL;DR

This paper completely solves two specific Diophantine equations involving Fibonacci and Jacobsthal sequences, identifying all cases where sums of two sequence terms equal a term of the other sequence.

Contribution

It provides a complete classification of solutions to the equations $J_n + J_m = F_a$ and $F_n + F_m = J_a$, which was previously unknown.

Findings

All solutions for $J_n + J_m = F_a$ are identified.

All solutions for $F_n + F_m = J_a$ are identified.

The paper characterizes when sums of two Fibonacci or Jacobsthal numbers equal a sequence term.

Abstract

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.