# Possible Solutions of the Diophantine equation x2+ky2=z2

## Abstract

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 $\left&space;(&space;\pm&space;\left&space;(&space;ka-b&space;\right&space;),\pm&space;2\sqrt{ab},\pm&space;\left&space;(&space;ka+b&space;\right&space;)&space;\right&space;)$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 $\left&space;(&space;\pm&space;\frac{ka-b}{2}&space;,\pm&space;\sqrt{ab},\pm&space;\frac{ka+b}{2}&space;\right&space;)$  where m is an integer, ab is an odd square number if y is odd, and $\left&space;(&space;\pm&space;\left&space;(&space;ka-b&space;\right&space;),\pm&space;2\sqrt{ab},\pm&space;\left&space;(&space;ka+b&space;\right&space;)&space;\right&space;)$ 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 $\left&space;(&space;\pm&space;\left&space;(&space;\frac{k}{4}a-b&space;\right&space;),\pm&space;\sqrt{ab},\pm&space;\left&space;(&space;\frac{k}{4}a+b&space;\right&space;)&space;\right&space;)$ where m is an integer, ab is a square number,

## Article Details

How to Cite
1.
Piyanut. Possible Solutions of the Diophantine equation x2+ky2=z2. Prog Appl Sci Tech. [Internet]. 2017 Oct. 25 [cited 2024 Aug. 12];7(2):200-5. Available from: https://ph02.tci-thaijo.org/index.php/past/article/view/243075
Section
Mathematics and Applied Statistics

## References

A. Wiles. Modula elliptic curves and Fermat last theorem. Annals of Mathematics. 142 (1995): 443-551.

N. Bruin. The Diophantine equation and . Compositio Mathematica. 118 (1999): 305-321.

M.A. Bennett. The equation . Journal de Théorie des Nombres de Bordeaux. 18 (2006): 315-321.

S. Abdelalim, H. Dyani. The solution of the Diophantine equation . International Journal of Algebra. 8 (2014): 729-732.

S. Abdelalim, H. Diany. Characterization of the solution of the Diophantine equation . Gulf Journal of Mathematics. 3 (2015): 1-4.