The Maximal Difference of Different Powers of an Element Modulo n

Abstract

In this paper, we investigate the maximal difference of integer powers of an element modulo $n$ . Let ${\left(a\right)}_n$ denote the integer $b$ with $1\le b\le n$ such that $a\equiv b\left(\mathrm{mod}n\right)$ for any integer $a$ . Using the bounds for exponential sums, we obtain a lower bound of the function $H_{m_1,m_2}\left(n\right):=\max \left\{\left|{\left(a^{m_1}\right)}_n-{\left(a^{m_2}\right)}_n\right|:1\le a\le n,\left(a,n\right)=1\right\},$ which gives $n-H_{m_1,m_2}\left(n\right)=O\left(n^{\left(3/4\right)+o\left(1\right)}\right)$ .