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

On the even power of the difference between the primitive root and its inverse modulo N and divisibility

设模 n≥ 3存在原根 ,对任一原根 1≤ a≤ n - 1且 ( a,n) =1 ,显然存在唯一的原根 1≤ a≤ n - 1使得 aa≡ 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 a与它的逆 a的差的偶数幂的分布性质及其整除性 ,并给出 ∑na=1| a-a|≤δna∈ Ak| a+ a( a - a) 2 h 的一个渐近公式 .

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≤≤n-1, so that a≡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 modulo n and its divisibility, and to give an asymptotic formula for ∑na=1｜a-｜≤δna∈Ak｜a+(a-) 2h