Integer solutions to $x^y - y^x = 1$

Question

The equation $x^y - y^x = 1$ has integer solutions $(2,1)$, $(3,2)$ and $(k, 0)$ (for any $k > 0$). Are there any others? Based on the graph (https://www.desmos.com/calculator/qyxoemixli, see below) it doesn't look like it, but is there a simple way to prove it?

Answer 1

This is a special case of Mihailescu's Theorem: https://en.wikipedia.org/wiki/Catalan%27s_conjecture.

I'm not sure if there's an easy way to prove this case though.