Pillai's conjecture revisited

We prove a generalization of an old conjecture of Pillai (now a theorem of Stroeker and Tijdeman) to the effect that the Diophantine equation 3^x−2^y=c has, for |c|>13, at most one solution in positive integers x and y. In fact, we show that if N and c are positive integers with N?2, then the equation |(N+1)^x−N^y|=c has at most one solution in positive integers x and y, unless (N,c)∈{(2,1),(2,5),(2,7),(2,13),(2,23),(3,13)}. Our proof uses the hypergeometric method of Thue and Siegel and avoids application of lower bounds for linear forms in logarithms of algebraic numbers.