On the Solutions of the Diophantine Equation $p^x +(p+4k)^y=z^2$ for Prime Pairs $p$ and $p+4k$

Authors

DOI:

https://doi.org/10.29020/nybg.ejpam.v14i2.3947

Keywords:

Diophantine equation, exponential Diophantine equation, nonlinear Diophantine equation, cousin primes, Legendre symbol

Abstract

In this paper, we solve the Diophantine equation px + (p + 4k)y = z2 in N0 for prime pairs (p, p+ 4k). First, we consider cousin primes p and p+ 4. Then we extend the study to solving px + (p + 4)y = z 2n, where n ∈ N\{1}. Furthermore, we solve the equation px + (p + 4k)y = z2 for k ≥ 2. As a result, we show that this equation has a unique solution (p, p + 4k, x, y, z) =(3, 11, 5, 2, 122) whenever x > 1 and y > 1. Finally, we show the finiteness of number of solutions in N.

Author Biographies

  • Renz Jimwel Silva Mina, University of the Philippines Baguio
    Mr. Renz Jimwel S. Mina is a full-time scholar pursuing Ph.D. Mathematics at the University of the Philippines Baguio. His research interests include exponential Diophantine equations, prime pairs, and elliptic curves.
  • 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.

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Published

2021-05-18

Issue

Vol. 14 No. 2: (April 2021)

Section

Nonlinear Analysis

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How to Cite

On the Solutions of the Diophantine Equation $p^x +(p+4k)^y=z^2$ for Prime Pairs $p$ and $p+4k$. (2021). European Journal of Pure and Applied Mathematics, 14(2), 471-479. https://doi.org/10.29020/nybg.ejpam.v14i2.3947