I am trying to find solutions to the system $p^x+q=q^y+r=r^z+p$ where $p,q,r$ are primes and $x,y,z$ are integers greater than $2$.
So far, I have only found that at most one among $x$, $y$, and $z$ can be even (factoring the two even powers), and if all of them are odd numbers, then there is only one residual class for $p$, $q$, and $r$ modulo $24$ (i.e., $a^x \equiv a \pmod{24}$ if $x$ is odd and $a$ coprime to $24$).\ Any references to similar problems or ideas to make progress would be appreciated!
