Positive integer solutions of the diophantine equation x 2 − L n xy + (−1) n y 2 = ±5 r

Abstract

In this paper, we consider the equation x 2 − L n xy + (−1) n y 2 = ±5 r and determine the values of n for which the equation has positive integer solutions x and y. Moreover, we give all positive integer solutions of the equation x 2 − L n xy + (−1) n y 2 = ±5 r when the equation has positive integer solutions.

Positive integer solutions of the diophantine equations $x^2 -5 F_n xy - 5(-1)^n y^2 = \pm 5^r$

“…Further details on Diophantine equations can be found in [7, 8, 17-19, 23, 24]. In [12], the authors considered the Diophantine equations…”