Non-existence of Positive Integer Solutions of the Diophantine Equation $p^x+(p+2q)^y=z^2$, where $p$, $q$ and $p+2q$ are Prime Numbers

Authors

  • Suton Tadee Thepsatri Rajabhat University
  • Apirat Siraworakun Thepsatri Rajabhat University

DOI:

https://doi.org/10.29020/nybg.ejpam.v16i2.4702

Keywords:

Diophantine Equation, Legendre symbol

Abstract

The Diophantine equation $p^x+(p+2q)^y=z^2$, where $p$, $q$ and $p+2q$ are prime numbers, is studied widely. Many authors give $q$ as an explicit prime number and investigate the positive integer solutions and some conditions for non-existence of positive integer solutions. In this work, we gather some conditions for odd prime numbers $p$ and $q$ for showing that the Diophantine equation $p^x+(p+2q)^y=z^2$ has no positive integer solution. Moreover, many examples of Diophantine equations with no positive integer solution are illustrated.

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Published

2023-04-30

Issue

Section

Nonlinear Analysis

How to Cite

Non-existence of Positive Integer Solutions of the Diophantine Equation $p^x+(p+2q)^y=z^2$, where $p$, $q$ and $p+2q$ are Prime Numbers. (2023). European Journal of Pure and Applied Mathematics, 16(2), 724-735. https://doi.org/10.29020/nybg.ejpam.v16i2.4702