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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?

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    $\begingroup$ I suspect this is a duplicate question. I think there is an elementary proof that the answer is negative. Further, I believe that between any two squares there are at most two powers, and that it is unknown if this happens infinitely many times. Gerhard "Look Up Guy and Pillai" Paseman, 2017.05.16. $\endgroup$
    – Gerhard Paseman
    Commented May 16, 2017 at 7:00

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According to the Wikipedia page on Catalan's conjecture, the problem you pose is open. (Look under the "generalization" heading.) A more general problem is Pillai's conjecture.

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