Diophantine equation $x^{y} - y^{x}=1$.

Question

I was thinking about the equation $x^{y} - y^{x}=1$ where $x,y \in \mathbb N$ and the solution $x=3$ and $y=2$ was easy to find. Also $x=2$ and $y=1$ is a solution.

I would like to know:

Is this Diophantine equation studied somewhere and are there any other solutions known?

Answer 1

For $x,y>1$ the only solution is $(x,y)=(3,2)$ by Mihăilescu's theorem. The remaining solutions are easily verified to be $(x,y)=(2,1)$ and $(x,y)=(x,0)$ for every $x>0$.