Possible Solutions of the Diophantine equation x2+ky2=z2 - Piyanut (Progress in Applied Science and Technology, 2017)
This paper is to identify the Diophantine equation x2+ky2=z2 where k, x, y and z are integers satisfies; case 1: k=4m+2, has no integer solution if y is odd, and have integer solutions (x, y, z) is <img title="( ± ( ka-b ),± 2ab,± ( ka+b ) )" src="https://latex.codecogs.com/gif.latex?&space;(&space;±&space;&space;(&space;ka-b&space;&space;),±&space;2ab,±&space;&space;(&space;ka+b&space;&space;)&space;&space;)">where m is an integer, ab is a square number if y is even, case 2: k=2m+1 , have integer solutions (x, y, z) is where m is an integer, ab is an odd square number if y is odd, and <img title="( ± ( ka-b ),± 2ab,± ( ka+b ) )" src="https://latex.codecogs.com/gif.latex?&space;(&space;±&space;&space;(&space;ka-b&space;&space;),±&space;2ab,±&space;&space;(&space;ka+b&space;&space;)&space;&space;)"> where m is an integer, ab is a square number if y is even, case 3: , k=4m have integer solutions (x, y, z) is <img title="( ± ( k4a-b ),± ab,± ( k4a+b ) )" src="https://latex.codecogs.com/gif.latex?&space;(&space;±&space;&space;(&space;k4a-b&space;&space;),±&space;ab,±&space;&space;(&space;k4a+b&space;&space;)&space;&space;)"> where m is an integer, ab is a square number,