About the power of numbers primes distribution

Question

Let $r>0$, $p\neq q$ two primes numbers and $A=\{(m,n)\in\mathbb N^2; |p^m-q^n|\leq r\}$.

Is it true that $A$ is a finite set?

Answer 1

Expanding on my comment, suppose we have an infinite increasing sequence such that $|p^m - q^n| \leq r$ for each $(n, m)$ in the sequence. $|p^m - q^n| = p^m |1 - e^{n\log q-m\log p}|$. We want this to be small, so from some point $|n\log q-m\log p|$ must be small enough that $|1 - e^{n\log q-m\log p}| > 0.9 |n\log q-m\log p|$. According to Baker's theorem, there exists a constant $C$ such that $|n\log q-m\log p| \geq \max(n, m) ^ {-C}$. This means $|1 - e^{n\log q-m\log p}|$ can decreases at most polynomially quickly in $m$, while $p^m$ increases exponentially, so $|p^m - q^n|$ must diverge to infinity.