Question 24
Denote by $d(n)$ the number of distinct positive divisors of a positive integer $n$ (including $1$ and $n$). Let $a \gt 1$ and $n \gt 0$ be integers such that $a^{n} + 1$ is a prime. Prove that
$$
d\left(a^{n} - 1\right) \geq n.
$$
$$
d\left(a^{n} - 1\right) \geq n.
$$