I was curious about the following diophantine equation $$x^xy^y=z^z$$ The equation seemed to have no positive integer solutions except for when $x$ or $y$ was $1$.
Though I attempted to solve the equation using prime factoring, I was able to proceed no further.
However, since I believe that I saw this question in a book, it is likely that this question is a duplicate or has been solved before. In that case I wish to apologize.
Any progress on the question would be appreciated.