关于Golomb猜想的一般化

On the generalization of Golomb's conjecture

设p为奇素数,c是任意与p互素的整数。那么Golomb猜想可以简单描述为对任意素数p≥3,存在模p的两个原根α,β,使得α+β≡c mod p。文中的主要目的是推广这一结果,即利用特征和的估计以及原根的判别性质证明更一般的结论:设p为充分大的素数,k为给定的正整数。对于任意给定的两两不同余的整数c1,c2,…,ck且(p,c1c2…ck)=1,一定存在模p的k+1个原根β1,β2,…,βk及α使得βi+α≡cimod p,i=1,2,…,k。显然当k=1时就是Golomb猜想。所以,该结果是Golomb猜想的进一步推广和延伸。

Let p be an odd prime, c be any integer with （p ,c） = 1. The Golomb＇s conjecture can be simply described as there exist two primitive roots α and β mod p such that α＋β≡c mod p. In this paper, we give a generalized result. That is, we will use the estimate for character sums mod p and the discriminant method of primitive roots mod p to prove the following general conclusion ： Let p be an odd prime large enough, k be any fixed positive integer. Then for any integers c1 , c2, …, ck with （c1c2…ck ,p） = 1, there exist k ＋ 1 primitive roots β1,β2,…,βk and α mod p so that βi＋α≡cimod p,i=1,2,…,k. It is clear that this result in fact is Golomb＇s conjecture, if k = 1. So the result is a generalization of Golomb＇s conjecture.