Question 147
Let $p$ be a given prime. For each prime $r$, we define the function as follows
$$
F(r) = \frac{(p^{r p} - 1)(p - 1)}{(p^r - 1)(p^p - 1)}
$$
1. Show that $F(r)$ is a positive integer for any prime $r \neq p$.
2. Show that $F(r)$ and $F(s)$ are coprime for any primes $r$ and $s$ such that $r \neq p$, $s \neq p$ and $r \neq s$.
3. Fix a prime $r \neq p$. Show that there is a prime divisor $q$ of $F(r)$ such that $p \mid q-1$ but $p^2 \nmid q-1$.
$$
F(r) = \frac{(p^{r p} - 1)(p - 1)}{(p^r - 1)(p^p - 1)}
$$
1. Show that $F(r)$ is a positive integer for any prime $r \neq p$.
2. Show that $F(r)$ and $F(s)$ are coprime for any primes $r$ and $s$ such that $r \neq p$, $s \neq p$ and $r \neq s$.
3. Fix a prime $r \neq p$. Show that there is a prime divisor $q$ of $F(r)$ such that $p \mid q-1$ but $p^2 \nmid q-1$.