On the Non-Linear Diophantine Equations $4^{x} - a^{y} = dz^{2}$ and $4^{x} + a^{y} = dz^{2}$

Authors

  • Suton Tadee
  • Umarin Pintoptang Naresuan University

Keywords:

Diophantine equation, integer solutions, congruence

Abstract

In this article, we study Diophantine equations $4^{x} - a^{y} = dz^{2}$ and $4^{x} + a^{y} = dz^{2}$ where $a, d, x, y,$ and $z$ are non-negative integers. Under some conditions of integers $a$, $d$ and by using congruence properties, we give all non-negative integer solutions of $4^{x} - a^{y} = dz^{2}$ and we show that $4^{x} + a^{y} = dz^{2}$ has no non-negative integer solution.

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Published

2023-09-30

How to Cite

Tadee, S., & Pintoptang, U. (2023). On the Non-Linear Diophantine Equations $4^{x} - a^{y} = dz^{2}$ and $4^{x} + a^{y} = dz^{2}$. Thai Journal of Mathematics, 21(3), 563–567. Retrieved from https://thaijmath2.in.cmu.ac.th/index.php/thaijmath/article/view/1527

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Section

Articles