ABSTRACT: This study determines all positive integer solutions of the Diophantine equations of the form x2 − x y −
y2 ± l x = 0 and x2 − 3x y + y2 ± l x = 0, where l is of the form m∏ni=1piai , pi is a prime congruent to 2 or 3 modulo 5,
and m is either 1 or 5. In addition, we also provide all positive integer solutions of x2 − x y − y2 ± l y = 0.
Algebra and Applications Research Unit, Department of Mathematics and Statistics, Faculty of Science,
Prince of Songkla University, Songkhla 90110 Thailand