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 < y^δ in x, y ∈ S for fixed δ ∈ (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.