The Maximal Difference of Different Powers of an Element Modulo n

In this paper, we investigate the maximal difference of integer powers of an element modulo n . Let a n denote the integer b with 1 ≤ b ≤ n such that a ≡ b mod   n for any integer a . Using the bounds for exponential sums, we obtain a lower bound of the function H m 1 , m 2 n : = max a m 1 n − a m 2 n : 1 ≤ a ≤ n , a , n = 1 , which gives n − H m 1 , m 2 n = O n 3 / 4 + o 1 .