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

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.