The The Diophantine Equations 8^x+p^y=z^3 and 8^x-p^y=z^3

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Published: Apr 1, 2023
Keywords:
Diophantine equation non-negative integer solution

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

Abstract

Let gif.latex?p  be a prime. In this paper, we show that all non-negative integer solutions of the Diophantine equation gif.latex?8^{x}+p^{y}=z^{3} are of the following form gif.latex?\left&space;(&space;p,x,y,z&space;\right&space;)=\left&space;(&space;3.2^{2n}+3.2^{n}+1,n,1,2^{n}+1\right&space;) , where gif.latex?n is a non-negative integer. All non-negative integer solutions of the Diophantine equation gif.latex?8^{x}-p^{y}=z^{3} are of the form gif.latex?\left&space;(&space;p,x,y,z&space;\right&space;), where 
gif.latex?\left&space;(&space;p,x,y,z&space;\right&space;)\in&space;\left&space;\{\left&space;(&space;2,n,3n,0&space;\right&space;),\left&space;(&space;p,0,0,0&space;\right&space;),\left&space;(&space;13,3,2,7&space;\right&space;),\left&space;(3.2^{2n}-3.2^{n}+1,n,1,2^{n}+1&space;,n,1,2^{n}-1&space;\right&space;)&space;\right&space;\}

Article Details

Issue
Vol. 15 No. 21 (2023): JANUARY 2023 - JUNE 2023
Section
บทความวิจัย
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This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

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