Let p,q be primes. Define the function \sigma:\mathbb{N} \to \mathbb{N} by \sigma(n) = k where k = \min \{a \in \mathbb{N} \mid a> n \log_p{q} \}. Define the function \tau:\mathbb{N} \to \mathbb{N} by \tau(n) = p^{\sigma(n)} - q^n. Does anyone have a suggestion for how I might approximate \tau(n)?