Question 193
A prime $p$ is called *nice prime* if there exists sequences of positive integers $(n_1, n_2, \dots, n_k)$ satisfying the following conditions for infinitely many positive integers $k$, but there does not exist such for $k = 1$.
1. For $i = 1, 2, \dots, k$, $n_i \ge \frac{i+1}{2}$.
2. For $i = 1, 2, \dots, k$, $p^{n_i} - 1$ is a multiple of $n_{i+1}$ and $\frac{p^{n_i} - 1}{n_{i+1}}$, $n_{i+1}$ are prime to each other. Set $n_{k+1} = n_1$.
Show that $2$ is not a nice prime, but any odd primes are nice primes.
1. For $i = 1, 2, \dots, k$, $n_i \ge \frac{i+1}{2}$.
2. For $i = 1, 2, \dots, k$, $p^{n_i} - 1$ is a multiple of $n_{i+1}$ and $\frac{p^{n_i} - 1}{n_{i+1}}$, $n_{i+1}$ are prime to each other. Set $n_{k+1} = n_1$.
Show that $2$ is not a nice prime, but any odd primes are nice primes.