Some Digraphs Attached with the Congruence x^5 ≡ 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, · · · , n − 1} and for which there is a directed edge from a ∈ h to b ∈ h if a5 ≡ b (mod n).we determine the number of the fixed points of γ(n). we also give the structure of γ(n) for n = 2^k and n = 5^k, 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 γ1(n) be the subdigraph inducedby the vertices which are coprime to n. we discuss when γ1(n) is regular