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

Article Sidebar

PDF
Published: Aug 30, 2022
Keywords:
Diophantine equation Integer solution Catalan’s conjecture

Main Article Content

Suton Tadee
Department of Mathematics, Faculty of Science and Technology, Thepsatri Rajabhat University, Lopburi 15000

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}^{+}


 

Article Details

Issue
Vol.30 No.4 (July - August 2022)
Section
Physical Sciences