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

TL;DR

This paper extends the analysis of the equation involving prime powers and an integer constant to nonnegative exponents, providing new elementary proofs and summarizing solution bounds for related equations.

Contribution

It generalizes previous results to nonnegative exponents and offers elementary proofs for key theorems, enhancing understanding of the equation's solutions.

Findings

At most two solutions in nonnegative integers, with exceptions.

Elementary proofs of Luca's results are provided.

Shorter proofs of Szalay's results are achieved.

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.