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
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.