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

Abstract

In this study, we consider the Diophantine equations given in the title and determine when these equations have positive integer solutions. Moreover, we find all positive integer solutions of them in terms of Fibonacci and Lucas numbers.

Infinitely many positive integer solutions of the quadratic Diophantine Equation $x^2-8B_nxy-2y^2=\pm 2^r$

“…Theorem 3.5. If n ≥ 1, then the equation P n = x 2 has positive solutions (n, x) = (1, 1) or (7,13).…”

“…Later, applying some properties of Fibonacci and Lucas numbers, they gave all positive integer solutions of (2) in terms of Fibonacci and Lucas numbers. In [13], Keskin, Karaatlı, and S ¸iar determined when the equations…”