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We improve earlier work on the title equation (where $p$ and $q$ are primesand $c$ is a positive integer) by allowing $x$ and $y$ to be zero as well aspositive. Earlier work on the title equation showed that, with listedexceptions, there are at most two solutions in positive integers $x$ and $y$,using elementary methods. Here we show that, with listed exceptions, there areat most two solutions in nonnegative integers $x$ and $y$, but the proofs aredependent on nonelementary work of Mignotte, Bennett, Luca, and Szalay. Inorder to provide some of our results with purely elementary proofs, we giveshort elementary proofs of the results of Luca, made possible by an elementarylemma 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 resultson the number of solutions to the generalized Pillai equation $(-1)^u r a^x +(-1)^v s b^y = c$ is also given.

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