For a prime $p$ and positive integer $n$, let $E(n,p)$ be the greatest $k$ such that $p^k \mid n$, and $E(n,p) = 0$ if $p \nmid n$. Let $E(n) = E(n, 2)$.
A number of years back, I proved the following results, and wondered how well-known they were (all variables are positive integers):
If $y \ge 3$ is even and $n$ is even then $E(y^n-1) = E(n) + \max(E(y-1), E(y+1))$.
If $y$ is even and $p \mid y-1$, where $p$ is an odd prime, then $E(y^n-1,p) = E(n,p)+E(y-1,p)$.