On the Diophantine Equations p^{2x}+q^{y}=z^{4} and p^{2x}-q^{y}=z^{4}  , Where p  and q  are Primes

Kulprapa   Srimud; Suton   Tadee

Authors

Kulprapa Srimud Department of Mathematics and Computer Sciences, Faculty of Science and Technology, Rajamangala University of Technology Thanyaburi
Suton Tadee Department of Mathematics, Faculty of Science and Technology, Thepsatri Rajabhat University

Keywords:

Diophantine equation , Catalan’s Conjecture

Abstract

In this paper, we study Diophantine equations $p^{2x}+q^{y}=z^{4}$ and $p^{2x}-q^{y}=z^{4}$ , where p and q are primes. We found that all non-negative integer solutions of the Diophantine equation $p^{2x}+q^{y}=z^{4}$ are of the following ( $p,q,x,y,z$ ) $\epsilon$ $\left\{(3,7,1,1,2)\right\}\cup$ $\left\{\left(p,2,1,log_{2}(p+1)+2,\sqrt{p+2}log_{2}(p+1),\sqrt{p+2}\epsilon\mathbb{Z}\right)\right\}$ $\cup$ $\left\{(2,17,3,1,3)\right\}$ and all non-negative integer solutions of the Diophantine equationare $p^{2x}-q^{y}=z^{4}$ of the following ( $p,q,x,y,z$ ) $\epsilon$ $\left\{\left(p,q,1,log_{q}(2p-1),\sqrt{p-1}\right)log_{q}(2p-1),\sqrt{p-1}\epsilon\mathbb{Z}\right\}\cup$ $\left\{\left(p,2,1,log_{2}(p-1)+2,\sqrt{p-2}log_{2}(p-1),\sqrt{p-2}\epsilon\mathbb{Z}\right)\right\}$ $\cup\left\{\left(p,q,0,0,0\right)\right\}\cup\left\{\left(p,p,u,2u,0\right)u\epsilon\mathbb{Z}^{+}\right\}$

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