The generalized Pillai equation ±rax±sby=c

In this paper we consider N, the number of solutions (x,y,u,v) to the equation u(−1)rax+v(−1)sby=c in positive integers x,y and integers u,v∈{0,1}, for given integers a>1, b>1, c>0, r>0 and s>0. We show that N⩽2 when gcd(ra,sb)=1, except for a finite number of cases that can be found in a finite number of steps. For arbitrary gcd(ra,sb) with (u,v)=(0,1), we show that N⩽3 with an infinite number of cases for which N=3.