On the Diophantine Equations $a^x+b^y+c^z=w^2$

Authors

  • Kittipong Laipaporn Department of Mathematics and Statistics, School of Science, Walailak University, Nakhon Si Thammarat , 80160
  • Saowapak Kaewchay Department of Mathematics and Statistics, School of Science, Walailak University, Nakhon Si Thammarat , 80160
  • Adisak Karnbanjong Department of Mathematics and Statistics, School of Science, Walailak University, Nakhon Si Thammarat , 80160

DOI:

https://doi.org/10.29020/nybg.ejpam.v16i4.4936

Keywords:

Exponential Diophantine Equation, Integers modulo $n$

Abstract

Over the past decade, exponential Diophantine equations of the form ax + by = wn have been studied as if they were a phenomenon. In particular, numerous articles have focused on the cases where n = 2 or n = 4 and 2 ≤ a, b ≤ 200. However, these articles are primarily concerned with determining whether the left-hand side of the equation needs to consist of more than two
exponentials. Therefore, in this article, we investigate the exponential Diophantine equation in the form ax + by + cz = w2, using only elementary tools related to modulo concepts. We present three theorems in which the variables a, b and c vary under certain conditions, and three additional theorems where the variable c is fixed at 7. Furthermore, if we restrict our parameters a, b and c
to 2 ≤ a ≤ b ≤ c ≤ 20, then 1,330 equations have been considered. Our results confirm that 135 of these equations have been fully clarified.

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Published

2023-10-30

How to Cite

Laipaporn, K., Kaewchay, S., & Karnbanjong, A. (2023). On the Diophantine Equations $a^x+b^y+c^z=w^2$. European Journal of Pure and Applied Mathematics, 16(4), 2066–2081. https://doi.org/10.29020/nybg.ejpam.v16i4.4936

Issue

Vol. 16 No. 4: (October 2023)

Section

Nonlinear Analysis

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