The expression $16^n + 4^n + 1$ is equivalent to the expression \( \frac{2^{p(n)} - 1}{2^{q(n)} - 1} \) for all positive integers \( n > 1 \), where \( p(n) \) and \( q(n) \) are functions and \( \frac{p(n)}{q(n)} \) is constant. Find \( p(2006) - q(2006) \).

Let's think step by step and output the final answer within \boxed{}.
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