How can I prove that for all $i, j \in \mathbb{N}$ there are only a finite number of solutions to $x^a + i + j = y^b + j = z^c$ with $a,b,c,x,y,z \in \mathbb{N}$ and $a,b,c \ge 2$? This is a weaker version of this generalization of Catalan's conjecture which would give a negative answer to this recent question.
I can see that the set of solutions has at most a certain density. I'm interested in learning new tools for solving this problem and also the linked question.