Loops in Digraphs of Lambert Mapping Modulo Prime Powers: Enumerations and Applications

Abstract

For an odd prime number p, and positive integers k and , we denote , a digraph for which is the set of vertices and there is a directed edge from u to v if , where . In this work, we study isolated and non-isolated fixed points (or loops) in digraphs arising from Discrete Lambert Mapping. It is shown that if , then all fixed points in are isolated. It is proved that the digraph has isolated fixed points only if . It has been characterized that has no cycles except fixed points if and only if either g is of order 2 or g is divisible by p . As an application of these loops, the solvability of the exponential congruence has been discussed.