Positive Integer Solutions of Certain Diophantine Equations Involving Lucas-Balancing numbers

TL;DR

This paper investigates the existence of positive integer solutions to specific Diophantine equations involving Lucas-Balancing numbers, expressing solutions through balancing, Pell, and Lucas-Pell numbers.

Contribution

It introduces new results on solutions of certain Diophantine equations involving Lucas-Balancing numbers, linking solutions to well-known number sequences.

Findings

Identifies conditions for positive solutions of the equations.

Expresses solutions in terms of balancing, Pell, and Lucas-Pell numbers.

Provides explicit solutions or criteria based on these sequences.

Abstract

In this article the Diophantine equations of the form $x^{2}-8C_{n}xy+ 16y^{2}=\pm2^{r}$, $x^{4}-8C_{n}x^{2}y+ 16y^{2}=\pm2^{r}$ and $x^{2}-8C_{n}xy^{2}+ 16y^{4}=\pm2^{r}$ are taken into consideration for the investigations of existence of any positive integer solutions. Furthermore, the solutions are manifested in terms of the balancing, Pell and Lucas-Pell numbers.