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

TL;DR

This paper completely characterizes solutions to specific Diophantine equations involving Lucas and Jacobsthal numbers, identifying all cases where sums of two such numbers equal another in the sequences.

Contribution

It provides a complete solution to the equations $L_n + L_m = J_k$ and $J_n + J_m = L_k$, using advanced number theory techniques.

Findings

All solutions to $L_n + L_m = J_k$ are identified.

All solutions to $J_n + J_m = L_k$ are identified.

The results are supported by Baker's theorem and reduction methods.

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.