On px−qy=c and related three term exponential Diophantine equations with prime bases

Using a theorem on linear forms in logarithms, we show that the equation px−2y=pu−2v has no solutions (p,x,y,u,v) with x≠u, where p is a positive prime and x,y,u, and v are positive integers, except for four specific cases, or unless p is a Wieferich prime greater than 1015. More generally, we obtain a similar result for px−qy=pu−qv>0 where q is a positive prime, q 1 mod 12. We solve a question of Edgar showing there is at most one solution (x,y) to px−qy=2h for positive primes p and q and positive integer h. Finally, we use elementary methods to show that, with a few explicitly listed exceptions, there are at most two solutions (x,y) to px±qy=c and at most two solutions (x,y,z) to px±qy±2z=0, for given positive primes p and q and integer c.