Pell and Pell-Lucas numbers as sums of two Jacobsthal numbers

TL;DR

This paper determines all solutions to equations expressing Pell and Pell-Lucas numbers as sums of two Jacobsthal numbers, using advanced logarithmic methods to solve these specific Diophantine equations.

Contribution

It provides a complete solution to the equations relating Pell, Pell-Lucas, and Jacobsthal numbers, a novel application of linear forms in logarithms in this context.

Findings

All solutions to the equations are explicitly characterized.

The equations have finitely many solutions, fully determined.

The method can be applied to similar Diophantine problems.

Abstract

We solve the two Diophantine equations $P_k=J_n+J_m$ and $Q_k=J_n+J_m$ where $\left\lbrace P_{k}\right\rbrace_{k\geq0}$, $\left\lbrace Q_{k}\right\rbrace_{k\geq0}$ and $\left\lbrace J_{k}\right\rbrace_{k\geq0}$ are the sequences of Pell numbers, Pell-Lucas numbers and Jacobsthal numbers, respectively. The main tool is the theory of linear forms in logarithms.