Pillaiʹs conjecture revisited Original Research Article

We prove a generalization of an old conjecture of Pillai (now a theorem of Stroeker and Tijdeman) to the effect that the Diophantine equation 3x−2y=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 Ngreater-or-equal, slanted2, then the equation (N+1)x−Ny=c has at most one solution in positive integers x and y, unless (N,c)set membership, variant{(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.