On the Diophantine Equation Mp^x + (Mq + 1)^y = z^2

William Sobredo Gayo, Jr.; Jerico Bravo Bacani

doi:10.29020/nybg.ejpam.v14i2.3948

Authors

William Sobredo Gayo, Jr. Don Mariano Marcos Memorial State University - North La Union Campus https://orcid.org/0000-0002-6448-7001
Jerico Bravo Bacani University of the Philippines Baguio http://orcid.org/0000-0002-3190-2408

DOI:

https://doi.org/10.29020/nybg.ejpam.v14i2.3948

Keywords:

Diophantine equation, Exponential Diophantine equation, Mersenne primes

Abstract

In this paper, we study and solve the exponential Diophantine equation of the form
Mxp + (Mq + 1)y = z2 for Mersenne primes Mp and Mq and non-negative integers x, y, and z. We use elementary methods, such as the factoring method and the modular arithmetic method, to prove our research results. Several illustrations are presented, as well as cases where solutions to the Diophantine equation do not exist.

Author Biographies

William Sobredo Gayo, Jr., Don Mariano Marcos Memorial State University - North La Union Campus

Mr. William S. Gayo, Jr. is a faculty of mathematics at the General Education Department, College of Arts and Sciences, Don Mariano Marcos Memorial State University - North La Union Campus, Bacnotan 2515, La Union, Philippines. He recently earned his master's degree in mathematics from the University of the Philippines Baguio.
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.

On the Diophantine Equation Mp^x + (Mq + 1)^y = z^2

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