Let $P = \{n^k: n,k\in\mathbb{N}\setminus\{0,1\}\}$ denote the set of powers. For any $n,r\in\mathbb{N}$ we set $B\_r(n)=\{m\in\mathbb{N}: |m-n| \leq r\}$. 


Is there a "global" constant $K\in\mathbb{N}$ such that the set $$\{q\in P: (B\_K(q)\setminus \{q\})\cap P \neq \emptyset\}$$ is infinite?



Bounded gaps between powers
According to the Wikipedia page on [Catalan's conjecture](https://en.wikipedia.org/wiki/Catalan%27s_conjecture), the problem you pose is open. (Look under the "generalization" heading.) A more general problem is Pillai's conjecture.


