On the Diophantine Equation p^x+(p+1)^y=z^2, where p is Prime

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Published: Jun 29, 2024
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Diophantine equation Non-negative integer solution Congruence

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Chantana Wannaphan
Department of Mathematics, Faculty of Science and Technology, Thepsatri Rajabhat University, Lopburi, Thailand
Suton Tadee
Department of Mathematics, Faculty of Science and Technology, Thepsatri Rajabhat University, Lopburi, Thailand

Abstract

In this paper, we study all non-negative integer solutions of the Diophantine equation  gif.latex?p^{x}+\left&space;(&space;p+1&space;\right&space;)^{y}=z^{2} , where gif.latex?p is a prime number. The research results showed that 1) If gif.latex?y=0, then the equation has only two solutions, namelyy gif.latex?\left&space;(&space;p,x,z&space;\right&space;)\in&space;\left&space;\{&space;\left&space;(&space;2,3,3&space;\right&space;)&space;,\left&space;(&space;3,1,2&space;\right&space;)\right&space;\},  2) if gif.latex?y=1, then the equation has only solutions in the form of gif.latex?\left&space;(&space;p,x,z&space;\right&space;)=\left&space;(&space;2,0,2&space;\right&space;) orgif.latex?\left&space;(&space;p,x,z&space;\right&space;)=\left&space;(&space;4n^{2}+4n-1,0,2n+1&space;\right&space;) , where gif.latex?n is a positive integer, 3) if gif.latex?y=2, then the equation has only two solutions, namely  gif.latex?\left&space;(&space;p,x,z&space;\right&space;)\in&space;\left&space;\{&space;\left&space;(&space;2,4,5&space;\right&space;)&space;,\left&space;(&space;3,2,5&space;\right&space;)\right&space;\}, 4) if gif.latex?p>3 and gif.latex?yis even, then the equation has no solution, and 5) if gif.latex?p=5, then the equation has only one solution. That is gif.latex?\left&space;(&space;x,y,z&space;\right&space;)=\left&space;(&space;4,3,29&space;\right&space;).

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How to Cite
Wannaphan, C., & Tadee, S. (2024). On the Diophantine Equation p^x+(p+1)^y=z^2, where p is Prime. Journal of Science Ladkrabang, 33(1), 1–7. Retrieved from https://li01.tci-thaijo.org/index.php/science_kmitl/article/view/258509
Issue
Vol. 33 No. 1 (2024): January - June 2024
Section
Research article
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This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

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