A Note on the Primitive Roots and the Golomb Conjecture

Abstract

In this paper, we use the elementary methods and the estimates for character sums to prove the following conclusion. Let p be a prime large enough. Then, for any positive integer n with p1/2+ɛ≤n<p, there must exist two primitive roots α and β modulo p with 1<α,β≤n−1 such that the equation n=α+β holds, where 0<ɛ<1/2 is a fixed positive number. In other words, n can be expressed as the exact sum of two primitive roots modulo p.