About certain prime numbers

TL;DR

This paper establishes a necessary condition for solutions to a specific Diophantine equation involving prime numbers, contributing to the understanding of prime-related equations.

Contribution

It introduces a new necessary condition for the existence of solutions to a prime-based Diophantine equation involving three distinct odd primes.

Findings

Identifies a necessary condition for solutions to the equation

Provides insights into the structure of solutions involving prime numbers

Advances understanding of prime number equations

Abstract

We give a necessary condition for the existence of solutions of the Diophantine equation $p=x^{q}+ry^{q},$ with $p$, $q$, $r$ distinct odd prime natural numbers.