BALTIC-WAY 1996 Problem 10

Denote by $d(n)$ the number of distinct positive divisors of a positive integer $n$ (including 1 and $n$ ). Let $a>1$ and $n>0$ be integers such that $a^{n}+1$ is a prime. Prove that

$$d\left(a^{n}-1\right) \geq n .$$