KMO 2010 Problem 6

An arbitrary prime $p$ is given. If an integer sequence $(n_1 , n_2 , \cdots , n_k )$ satisfying the conditions

• For all $i= 1, 2, \cdots , k$ , $n_i \geq \frac{p+1}{2}$
• For all $i= 1, 2, \cdots , k$ , $p^{n_i} - 1$ is divisible by $n_{i+1}$ , and $\frac{p^{n_i} - 1}{n_{i+1}}$ is coprime to $n_{i+1}$ . Let $n_{k+1} = n_1$ . exists not for $k=1$ , but exists for some $k \geq 2$ , then call the prime a good prime. Prove that a prime is good iff it is not $2$ .