Non-existence of integer solutions for the Diophantine equation \(p^x + p^y + n^z = w^2\), where p is an odd prime number and n is a positive integer

Non-existence of integer solutions for the Diophantine equation \(p^x + p^y + n^z = w^2\), where p is an odd prime number and n is a positive integer

Authors

Thepsatri Rajabhat University - Department of Mathematics
Thepsatri Rajabhat University - Department of Mathematics

Keywords:

Diophantine equation, Legendre symbol, congruence

Abstract

In this research, we investigate some conditions for the non-existence of integer solutions of the Diophantine equation \(p^x + p^y + n^z = w^2\), where p is an odd prime number and n is a positive integer. Moreover, numerous examples to illustrate these cases are provided.

Cover of n°53 Italian Journal of Pure and Applied Mathematics

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Published

2025-05-19

How to Cite

Suton, & Apirat. (2025). Non-existence of integer solutions for the Diophantine equation \(p^x + p^y + n^z = w^2\), where p is an odd prime number and n is a positive integer. Italian Journal of Pure and Applied Mathematics, 53, 151–165. Retrieved from https://journals.uniurb.it/index.php/ijpam/article/view/5717