On the Diophantine equation x2−kxy+y2+lx=0,l∈{1,2,4}

Abstract

We prove that the Diophantine equation x 2 − k x y + y 2 + l x = 0 , l ∈ { 1 , 2 , 4 } has an infinite number of positive integer solutions x and y if and only if ( k , l ) = ( 3 , 1 ) , ( 3 , 2 ) , ( 4 , 2 ) , ( 3 , 4 ) , ( 4 , 4 ) , ( 6 , 4 ) . Furthermore, we prove that the Diophantine equation x 2 − k x y + y 2 + x = 0 has infinitely many integer solutions x and y if and only if k ≠ 0 , ± 1 , which answers a problem in Marlewski and Marzycki (2004) [1] .