On ax-by=c

Aaron Hersehefield has recently proved the very interesting result that

2^x-3^y=c                                               (1)

has got at most one solution when c is large. Further, by applying the following theorem of Siegel, namely,

axⁿ-byⁿ=k (fixed n ≥ 3)

has at most one solution if |ab| is sufficiently large, he proves that

a^x-b^y=c                                                (2)

has at most 9 solutions.