A note on the Diophantine equation |a^x-b^y|=c - He, Bo; Togbé, Alain; Yang, Shichun (Mathematica Scandinavica, 2010)
Let a,b, and c be positive integers. We show that if (a,b) =(N^k-1,N), where N,k≥ 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 ± 1)^x - N^y|=c has more than one positive integer solutions (x,y).