Application of the p-Adic Topology on ℤ to the Problem of Finding Solutions in Integers of an Implicit Linear Difference Equation

Abstract

We study solutions in integers of an implicit linear inhomogeneous first order difference equation bx_n+1 = ax_n + f_n. Based on the p-adic topology on the ring of integers, we obtain a criterion for the existence of solutions and show that for a = 1 a typical (in the natural topological sense) equation has no integer solutions.