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In this work, the author shows a sufficient and necessary condition for aninteger 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. Aconstructive method of this type of integers is explained with details andexamples. Links between the main result and known ideas such as Termat's lasttheorem, Goormaghtigh conjecture and Mersenne numbers are discussed. otherrelated ideas, examples and applications are provides.

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