On the Diophantine Equation a^x+(a+2)^y=z^2

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Published: Feb 15, 2024
DOI: https://doi.org/10.14456/kkuscij.2024.4
Keywords:
Diophantine Equation Non-negative Integer Solution Congruence Mihăilescu’s Theorem

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

Abstract

In this paper, we investigated the solutions of the Diophantine equation gif.latex?a^{x}+(a+2)^{y}=z^{2}, where gif.latex?a is a positive integer and gif.latex?x,y,z are non-negative integers. Let gif.latex?S be the set of non-negative integer solutions gif.latex?(x,y,z) of the equation. The results showed that 1) if gif.latex?a is a prime number with gif.latex?a\equiv&space;5\left&space;(&space;mod8&space;\right&space;), then gif.latex?S&space;\equiv&space;\left&space;\{&space;(0,1,\sqrt{a+3})&space;\right&space;\}, where gif.latex?\sqrt{a+3} is an integer, otherwise gif.latex?S=\phi.  2) If gif.latex?a+2 is a prime number and gif.latex?x is even and the equation has a solution, then gif.latex?y=1 and gif.latex?z=2.  3) Let gif.latex?p be a prime number such that gif.latex?p\equiv&space;5,7(mod8) and gif.latex?a\equiv&space;-2(modp). Then gif.latex?S=\left&space;\{&space;(1,0,\sqrt{a+1})&space;\right&space;\}, where gif.latex?\sqrt{a+1} is an integer, otherwise gif.latex?S=\phi, when it satisfies one of the following cases: case 1 gif.latex?a\equiv&space;3\left&space;(&space;mod4&space;\right&space;) or case 2 there exists a prime number gif.latex?q such that gif.latex?q\equiv&space;3,5(mod8) and gif.latex?a\equiv&space;-1\left&space;(&space;modq&space;\right&space;).

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How to Cite
Tadee, S. (2024). On the Diophantine Equation a^x+(a+2)^y=z^2. KKU Science Journal, 52(1), 39–46. https://doi.org/10.14456/kkuscij.2024.4
Issue
Vol. 52 No. 1 (2024): January - April 2024
Section
Research Articles
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This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

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