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="\left ( ± \left ( ka-b \right ),± 2\sqrt{ab},± \left ( ka+b \right ) \right )" src="https://latex.codecogs.com/gif.latex?\left&space;(&space;±&space;\left&space;(&space;ka-b&space;\right&space;),±&space;2\sqrt{ab},±&space;\left&space;(&space;ka+b&space;\right&space;)&space;\right&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 <img title="\left ( ± \frac{ka-b}{2} ,± \sqrt{ab},± \frac{ka+b}{2} \right )" src="https://latex.codecogs.com/gif.latex?\left&space;(&space;±&space;\frac{ka-b}{2}&space;,±&space;\sqrt{ab},±&space;\frac{ka+b}{2}&space;\right&space;)"> where m is an integer, ab is an odd square number if y is odd, and <img title="\left ( ± \left ( ka-b \right ),± 2\sqrt{ab},± \left ( ka+b \right ) \right )" src="https://latex.codecogs.com/gif.latex?\left&space;(&space;±&space;\left&space;(&space;ka-b&space;\right&space;),±&space;2\sqrt{ab},±&space;\left&space;(&space;ka+b&space;\right&space;)&space;\right&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="\left ( ± \left ( \frac{k}{4}a-b \right ),± \sqrt{ab},± \left ( \frac{k}{4}a+b \right ) \right )" src="https://latex.codecogs.com/gif.latex?\left&space;(&space;±&space;\left&space;(&space;\frac{k}{4}a-b&space;\right&space;),±&space;\sqrt{ab},±&space;\left&space;(&space;\frac{k}{4}a+b&space;\right&space;)&space;\right&space;)"> where m is an integer, ab is a square number,