A note on the Diophantine equation $|a^x-b^y|=c$

Let a,b,and cbe positive integers. We show that if (a,b) =(N^k-1,N) where N,k\geq 2 then there is at most one positive integer solution (x,y)to the exponential Diophantine equation |a^x-b^y|=c unless (N,k)=(2,2) Combining this with results of Bennett [3] and the first author [6], we stated all cases for which the equation |(N^k \pm 1)^x - N^y|=chas more than one positive integer solutions...