  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.
