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

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.