Find all $x,y\in \mathbb{N}$, such that $ x^{y}=y^{y-x} $.
I have found the solutions $(1,1)$ and $(2,4)$. I've been trying to prove that there aren't any more. I tried splitting the cases by parity, but nothing interesting seemed to come out.
I also tried rearranging it into:
$x=y^{\frac{y-x}{y}}$.
Is there any way to proceed with this problem?
Thank you very much