On the Solutions of the Diophantine Equations (p-1)^x+2.p^y=z^2 and (p-1)^x-2.p^y=z^2

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Published: Dec 27, 2024
Keywords:
Diophantine equation Quadratic residue Legendre symbol

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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 number and gif.latex?x,y,z be non-negative integers. We show that the Diophantine equation gif.latex?\left&space;(&space;p-1&space;\right&space;)^{x}+2\cdot&space;p^{y}=z^{2} has all non-negative integer solutions, which are gif.latex?\left&space;(&space;p,x,y,z&space;\right&space;)=\left&space;(&space;3,1,0,2&space;\right&space;) and gif.latex?\left&space;(&space;p,x,y,z&space;\right&space;)=\left&space;(&space;2,t,2,3&space;\right&space;), where gif.latex?t is a non-negative integer. The Diophantine equation gif.latex?\left&space;(&space;p-1&space;\right&space;)^{x}-2\cdot&space;p^{y}=z^{2} has all non-negative integer solutions, which are gif.latex?\left&space;(&space;p,x,y,z&space;\right&space;)=\left&space;(&space;3,1,0,0\right&space;) and gif.latex?\left&space;(&space;p,x,y,z&space;\right&space;)=\left&space;(&space;p,1,0,\sqrt{p-3}&space;\right&space;), where gif.latex?p\equiv&space;7\left&space;(&space;mod12&space;\right&space;) such that gif.latex?\sqrt{p-3} is an integer.

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How to Cite
Tadee, S. (2024). On the Solutions of the Diophantine Equations (p-1)^x+2.p^y=z^2 and (p-1)^x-2.p^y=z^2. Journal of Science and Technology CRRU, 3(2), 15–22. retrieved from https://li01.tci-thaijo.org/index.php/jstcrru/article/view/263842
Issue
Vol. 3 No. 2 (2024): July - December
Section
Research article
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