On the Diophantine Equation�x^2-kxy+ly^2+my=0

Abstract

In this study, we determine all values of positive integer k for which the equations x^2-kxy+ky^2+2^ny=0, n\in {3,4,5,6,7} and x^2-kxy+ky^2+3^ny=0, n\in {1,2,3} have infinitely many positive integer solutions x and y. Furthermore, we investigate the values of positive integer k such that the Diophantine equation x^2-kxy+y^2+mkx=0 where m\in {1,2,3,5}, has infinitely many positive integer solutions x and y.???????????????? ????????????????????????????????? k ??????????????????????????????? x^2-kxy+ky^2+2^ny=0, n\in {3,4,5,6,7} ??? x^2-kxy+ky^2+3^ny=0, n\in {1,2,3} ???????????????????? x ??? y ???????????????? ????????? ????????????????? k ???????????????????????? x^2-kxy+y^2+mkx=0 ?????? m\in {1,2,3,5} ???????????????????? x ??? y ????????????????