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

Abstract

In this article the Diophantine equations of the form $x^2-8C_{n}xy+16y^2=\pm 2^r$, $x^4-8C_n{x}^{2}y+16y^2=\pm 2^r$ and $x^2-8C_{n}x{y}^2+16y^4=\pm 2^r$ 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.

There is some technical problems and issues with some of the contents of the paper, so I want to withdraw the manuscript