I'm taking a class on number theory and this is one of the problems my professor gave.
How many ordered integer triples $(x,y,z)$ are there such that $x^y-y^x=2017\times z$, where $x,y$ are less than $2017^2$?
My thought has been to take it$\mod 2017$, but I have not gotten far.
Let $p$ be prime; we can count the number of solutions to $x^y=y^x$ with $x,y \leq p^2$ and $p$ not dividing $x,y$ as follows:
Write $x=g^{a_1}$ mod $p$ and $x=a_2$ mod $(p-1)$; $y=g^{b_1}$ mod $p$ and $y=b_2$ mod $(p-1)$, where $g$ is a primitive root of $p$. Then the given equation reduces to $a_1b_2=a_2b_1$ mod $(p-1)$. Now count the number of such solutions by counting the solutions modulo each prime power factor of $(p-1)$.