Question 9
Let $S$ be the set of rational numbers of the form
$$
\frac{(a_1^2 + a_1 - 1)(a_2^2 + a_2 - 1) \cdots (a_n^2 + a_n - 1)}{(b_1^2 + b_1 - 1)(b_2^2 + b_2 - 1) \cdots (b_n^2 + b_n - 1)},
$$
where $n, a_1, a_2, \dots, a_n, b_1, b_2, \dots, b_n$ run through the positive integers. Show that $S$ contains infinitely many primes.
$$
\frac{(a_1^2 + a_1 - 1)(a_2^2 + a_2 - 1) \cdots (a_n^2 + a_n - 1)}{(b_1^2 + b_1 - 1)(b_2^2 + b_2 - 1) \cdots (b_n^2 + b_n - 1)},
$$
where $n, a_1, a_2, \dots, a_n, b_1, b_2, \dots, b_n$ run through the positive integers. Show that $S$ contains infinitely many primes.