A Note on the Primitive Roots and the Golomb Conjecture

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 p 1 / 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 .