  In this paper, we develop a new method for finding all perfect powers which
can be expressed as the sum of two rational S-units, where S is a finite set of
primes. Our approach is based upon the modularity of Galois representations
and, for the most part, does not require lower bounds for linear forms in
logarithms. Its main virtue is that it enables to carry out such a program
explicitly, at least for certain small sets of primes S; we do so for S = {2,
3} and S = {3, 5, 7}.
