## Differences of prime powers

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 $\tau(n)$?