RO-TST 2013 Problem 7

Let $S$ be the set of all rational numbers expressible in the form $$\frac{(a_1^2+a_1-1)(a_2^2+a_2-1)\ldots (a_n^2+a_n-1)}{(b_1^2+b_1-1)(b_2^2+b_2-1)\ldots (b_n^2+b_n-1)}$$ for some positive integers $n, a_1, a_2 ,\ldots, a_n, b_1, b_2, \ldots, b_n$ . Prove that there is an infinite number of primes in $S$ .