Positive integer solutions of certain diophantine equations

Abstract

In this study, the diophantine equations \(x^2 -32B_nxy-32y^2 =\pm 32^{r}\), \(x^4 -32B_nxy-32y^2 =\pm 32^{r}\) and \(x^2 -32B_nxy-32y^4 =\pm 32^{r}\) are considered and determined when these equations have positive integer solutions. Moreover, all positive integer solutions of these diophantine equations in terms of balancing and Lucas-balancing numbers are also found out.