Some Digraphs Attached with the Congruence x5 ≡ y (mod n)

In this paper, we associate to each natural number n a digraph Γ(n) whose set of vertices is H = {0, 1, · · · , n5− 1} and for which there is a directed edge from a ∈ H to b ∈ H if a ≡ b (mod n). We determine the number of the fixed points of Γ(n). We also give the and n = 5 , where k is a natural number. Making use of the Carmichael’s Theorem, we present a simple condition for the existence of cycles in Γ(n). Let Γ (n) be the subdigraph induced 1 by the vertices which are coprime to n. We discuss when Γ1(n) is regular