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On the Diophantine Equation $x^2 - kxy + ky^2 + ly=0, \, l \in \{1,2,4,8\}$

Volume 14, Number 1 (2012), 24 - 29

On the Diophantine Equation $x^2 - kxy + ky^2 + ly=0, \, l \in \{1,2,4,8\}$

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Abstract

In this study, we consider the Diophantine equation $x^{2}-kxy+ky^{2}+ly=0,$ with $l\in \left\{ 1,2,4,8\right\} $ and determine the values of $k$ when the equation has infinitely many positive integer solutions $x$ and $y.$