Problem: Find all positive integer solutions to the equation $a^a=a^b+b^a$.
Attempt at a solution: I first acknowledged that $a > b$, since if $a \le b$, then $a^a=a^b+b^a\ge a^a+a^a=2a^a$, which isn't true for any positive integer $a$. Then I used $a-b=d$, which, of course means $a=b+d$, and I substituted that into the original equation. I got $(d+q)^d[(d+q)^d-1]=q^dq^q$. I'm stuck here. Any help would be appreciated.