Question 7
Find all positive integers $n$ such that there exist non-constant polynomials with integer coefficients $f_1(x), \dots, f_n(x)$ (not necessarily distinct) and $g(x)$ such that
$$
1 + \prod_{k=1}^{n} (f_k^2(x) - 1) = (x^2 + 2013)^2 g^2(x).
$$
$$
1 + \prod_{k=1}^{n} (f_k^2(x) - 1) = (x^2 + 2013)^2 g^2(x).
$$