On the Diophantine Equation $p^x + (p + 5k)^y = z^2$

Authors

DOI:

https://doi.org/10.29020/nybg.ejpam.v18i3.6593

Keywords:

Exponential Diophantine equation, prime pairs, Mihailescu's theorem

Abstract

With the use of modular arithmetic and other fundamental number theoretic methods, as well as the concepts of floor function and the principle of mathematical induction, this study searches for possible nonnegative integer solutions of exponential Diophantine equations of the form $p^x + (p+5k)^y = z^2$, where $k \in \mathbb{N}$. Results are obtained for the following cases: 

\begin{itemize}

\item [a)] when $p =2$; or

\item [b)] when $p$ and $p + 5k$ are prime pairs.

In addition, the study is limited only to solutions where $x$ and $y$ are not both greater than 1.

Author Biographies

  • Jerico Bravo Bacani, University of the Philippines Baguio
    Dr. Jerico B. Bacani is a professor of mathematics at the University of the Philippines Baguio. He has served as Department Chairman for nine academic years (AYs 2006 - 2009, AYs 2014-2020). His research interests include Analysis and Number Theory.
  • Merlyn C. Avenilla, University of the Philippines Baguio

    Merlyn C. Avenilla is a graduate of the University of the Philippines, and a member of the Number Theory Research Group at the Department of Mathematics and Computer Science. As an early-career mathematician, her research interests include number theory, with a particular focus on Diophantine equations. She is currently gaining foundational experience in conducting studies in this area.

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Published

2025-08-01

Issue

Vol. 18 No. 3: (July 2025)

Section

Number Theory

License

Copyright (c) 2025 Jerico Bravo Bacani, Merlyn C. Avenilla

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How to Cite

On the Diophantine Equation $p^x + (p + 5k)^y = z^2$. (2025). European Journal of Pure and Applied Mathematics, 18(3), 6593. https://doi.org/10.29020/nybg.ejpam.v18i3.6593