Are there positive integers $s$ and $t$ such that $a^t - b^s = \pm 1$?

Question

Let $a$, $b$ two coprimes positive integers. The question is: are there positive integers $s$ and $t$ such that $a^t - b^s = \pm 1$?

Equivalently, if we define $d(a,b) = \min_{t,s > 0}( | a^t - b^s | )$ like a "distance", the question would be if $d(a,b) = 1$ when $a$, $b$ are coprimes.

It is clear this is false when $a$, $b$ aren't coprimes but I don't know how to work the other case

Answer 1

$3^2-2^3=1$

This is the only such nontrivial pair.

https://en.wikipedia.org/wiki/Catalan%27s_conjecture