关于模N的原根与它的逆的差的偶数幂的分布及其整除性

摘要

Let modulo n ≥ 3 contain a primitive root, and for each primitive root 1 ≤ a ≤ n - 1 with (a ,n)＝ 1 ,it is clear that there exists one and only one primitive root 1 ≤ α ≤ n - 1, so that αα ≡ 1 (modn) and iteger 1 ≤ k ＜ n with (k,n) ＝ 1. The main purpose of this paper, is to study the distribution properties of the even power of the difference between the primitive root and its inverse α modulo n and its divisibility, and to give an asymptotic formula for nΣa=1 |a-a|≤δn aэA k|a+a (a-a)2h.%设模n≥3存在原根，对任一原根1≤α≤n-1且(α，n)＝1，显然存在唯一的原根1≤α≤n-1使得αα≡1(modn)，对给定的正整数1≤k＜n且(k，n)＝1，本文主要目的是研究模n的原根α与它的逆α的差的偶数幂的分布性质及其整除性，并给出的一个渐近公式.