On the Diophantine Equations p^{2x}+q^{2y}=z^2 and p^{2x}-q^{2y}=z^2

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Suton Tadee

Abstract

Let gif.latex?p,q be prime numbers. In this paper, we prove that the Diophantine equation   gif.latex?p^{2x}+q^{2y}=z^{2}has a unique positive integer solution, that is gif.latex?\left&space;(&space;p,q,x,y,z&space;\right&space;)=\left&space;(&space;3,2,1,2,5&space;\right&space;) . We also show that all positive integer solutions of the Diophantine equation gif.latex?p^{2x}-q^{2y}=z^{2}are of the following gif.latex?\left&space;(&space;p,q,x,y,z&space;\right&space;)=\left&space;(4^{n-1}+1,2,1,n,4^{n-1}-1&space;\right&space;)&space;:&space;n&space;\in&space;Z^{+}-\left&space;\{&space;1&space;\right&space;\}&space;\cup&space;{(p,q,m,log_{q})}&space;\sqrt{2p^{m}}-1,p^{m}-1&space;:&space;m,log_{q}\sqrt{2p^{m}-1\in&space;}&space;\mathbb&space;{Z}^{+}


 

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