Possible Solutions of the Diophantine equation x2+ky2=z2

Piyanut Puangjumpa

Abstract


This paper is to identify the Diophantine equation x2+ky2=z2 where k,x,y and z are integers satisfies, case1, k=4m+2, has no integer solution if y is odd, and have integer solutions (x,y,z) is ((4m+2)a-b,2ab,(4m+2)a+b) where m is an integer, ab is a square number if y is even, case 2, k~=4m+2, have integer solutions (x,y,z) is ((ka-b)/2,ab,(ka+b)/2) where m is an integer, ab is an odd square number if y is odd, and (ka-b,2ab,ka+b) where m is an integer, ab is a square number if y is even.

Keywords


Diophantine equation, Congruence, Integer solutions, Divisibility

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Progress in Applied Science and Technology

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