Bounding super exponential functions with factorial functions

Question

I want to show that there exists polynomials $q(n)$, $p(n)$ with integer coefficeints such that $$ (q(n)!)^{p(n)} > 2^{2^n} $$ for all $n \in \mathbb{N}^+$. Intuitively this inequality seems to make since, and rought numerical simulations I have done suggest that it holds, but I cannot figure out a proof. Does anyone have any ideas as to whether this is true or false?

Answer 1

Let's take logarithms of both sides, we then want to have $p(n) \cdot \log_2 q(n)! > 2^n$.

We can use very rough estimation, $k! < k^k$, to get that left side is less than $p(n) \cdot q(n)\cdot \log_2 q(n)$. Which is less than $p(n) \cdot q(n) \cdot cn$ - polynomial. Given that right side is exponential in $n$, for all large enough $n$ the inequality is false.