Abstract
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 .
Highlights
Let n ≥ 3 be an integer and 1 ≤ a ≤ n be an integer with (a, n) 1
We investigate the maximal difference of integer powers of an element modulo n
We know that there exists a unique integer c such that ac ≡ 1(mod n), where c is called the inverse of a modulo n with 1 ≤ c < n
Summary
Let n ≥ 3 be an integer and 1 ≤ a ≤ n be an integer with (a, n) 1. 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. We know that there exists a unique integer c such that ac ≡ 1(mod n), where c is called the inverse of a modulo n with 1 ≤ c < n. Zhang [1] is the first person to explicitly study the distribution between an integer and its inverse modulo n, proving that n
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