Solving exponential diophantine equations using lattice basis reduction algorithms

Let S be the set of all positive integers with prime divisors from a fixed finite set of primes. Algorithms are given for solving the diophantine inequality 0<x - y <yd in x, y ¿ S for fixed d ¿ (0, 1), and for the diophantine equation x + y = z in x, y, z ¿ S. The method is based on multi-dimensional diophantine approximation, in the real and p-adic case, respectively. The main computational tool is the L3-Basis Reduction Algorithm. Elaborate examples are presented.