In the last post we conjectured that given fixed integers with , the equation

has only finitely many solutions in integers. Let us prove this conjecture.

If has no solution we are done. Otherwise suppose that is a solution. Assume the contrary, and let denote the infinite set of solutions . For any , we have

.

So, if has infinitely many solutions, then writing

gives that for infinitely many . So by Theorem 1.1 in this paper (or this one),

for some positive integers . So (exercise). But then divides for all , implying is bounded over all . But then, by , must also be bounded over all . Thus is finite, a contradiction.