The equation $|p^x \pm q^y| = c$ in nonnegative $x$, $y$

Abstract

We improve earlier work on the title equation (where $p$ and $q$ are primes and $c$ is a positive integer) by allowing $x$ and $y$ to be zero as well as positive. Earlier work on the title equation showed that, with listed exceptions, there are at most two solutions in positive integers $x$ and $y$, using elementary methods. Here we show that, with listed exceptions, there are at most two solutions in nonnegative integers $x$ and $y$, but the proofs are dependent on nonelementary work of Mignotte, Bennett, Luca, and Szalay. In order to provide some of our results with purely elementary proofs, we give short elementary proofs of the results of Luca, made possible by an elementary lemma which also has an application to the familiar equation $x^2 + C = y^n$. We also give shorter simpler proofs of Szalay's results. A summary of results on the number of solutions to the generalized Pillai equation $(-1)^u r a^x + (-1)^v s b^y = c$ is also given.