, 2025

Let $a \geq 3$ be an integer, and define $f(n) = a^n - 1$ for every positive integer $n$. Denote by $f^{(k)}$ the $k$-iterate of $f$, that is, $f^{(1)}(n) = f(n)$ and $f^{(k+1)}(n)=f(f^{(k)}(n))$ for $k \geq 1$.