# On the prime decomposition of integers of the form $(z^n-y^n)/(z-y)$

## TL;DR

This paper establishes a necessary and sufficient condition for integers of the form rac{zn - y^n}{z - y} to be divisible by perfect powers, connecting classical number theory problems with new divisibility criteria.

## Contribution

It provides a novel characterization of divisibility by perfect powers for a class of integers related to classical conjectures and theorems.

## Key findings

- Derived a necessary and sufficient condition for divisibility by perfect powers.
- Presented a constructive method with examples for generating such integers.
- Explored links to Fermat's Last Theorem, Mersenne numbers, and the Goormaghtigh conjecture.

## Abstract

In this work, the author shows a sufficient and necessary condition for an integer of the form $(zn-y^n)/(z-y)$ to be divisible by some perfect $mth$ power $p$, where $p$ is an odd prime and $m$ is a positive integer. A constructive method of this type of integers is explained with details and examples. Links between the main result and known ideas such as Termat's last theorem, Goormaghtigh conjecture and Mersenne numbers are discussed. other related ideas, examples and applications are provides.