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. +en, for any positive integer n with p(1/2)+ε ≤ 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< ε< (1/2) is a fixed positive number. In other words, n can be expressed as the exact sum of two primitive roots modulo p.


Introduction
Let p be a prime and F q be a finite field of q (� p h , h ≥ 1) elements with characteristic p. e Golomb conjecture (see [1]) can be summarized as follows: for any nonzero element c ∈ F q , there exist two primitive elements α and β ∈ F q such that the equation α + β � c holds.
If we assume that A(p) denotes the set of all primitive roots g modulo p with 1 ≤ g ≤ p − 1, then the Golomb conjecture in a reduced residue system modulo p can be described as that, for any integer 1 ≤ n ≤ p − 1, there exist two primitive roots α and β ∈ A(p) such that the congruence α + β ≡ nmodp holds.
A natural question is whether for a fixed 1 < n < p, there are two primitive roots 1 < α, β ≤ n − 1 of p such that α + β � n?. (1) Of course, for some positive integers n, equation (1) has no solutions. For example, n � 1, n � 2 and 3. So, we think that the problem in (1) is meaningful, and it is also closely related to the minimum primitive root modulo p.
On the other hand, we also want to know how large n is (relative to p), so that equation (1) must have a solution.
For the sake of convenience, for any odd prime p and integer 1 < n ≤ p − 1, let S(n; p) denote the number of all solutions of the equation α + β � n, where α and β are two primitive roots modulo p with 1 < α, β ≤ n − 1.
In this paper, we shall use the elementary methods and the estimates for character sums to study the asymptotic properties of S(n; p) and prove the following. Theorem 1. Let p be an odd prime. en, for any integer 1 < n ≤ p − 1, we have the asymptotic formula: where, as usual, ϕ(k) denotes the Euler function and ω(k) denotes the number of all distinct prime divisors of k.
It is clear that, for any positive number 0 < ε < (1/2), if prime p is large enough, then our theorem is nontrivial for at is, the main term is much big than the error term in our theorem. So, from our theorem, we may immediately deduce the following: Corollary 1. Let p be an odd prime large enough, 0 < ε < (1/2) be a fixed positive number. en, for any positive integer n with p (1/2)+ε ≤ n ≤ p − 1, there must exist two primitive roots α and β modulo p such that Note: first, the conclusion in our theorem can also be generalized. at is, let p be an odd prime and k be a fixed positive integer. For any integers 1 < n 1 < n 2 < n 3 < · · · < n k < p, if S(n 1 , n 2 , . . . , n k ; p) denotes the number of all solutions of the equations n 1 � α + β 1 , n 2 � α + β 2 , . . . , n k � α + β k , where α and all β i (i � 1, 2, . . . , k) are the primitive roots modulo p, then we have the following asymptotic formula: If 0 < ε < (1/2) and p (1/2)+ε < n 1 < p, then this asymptotic formula is nontrivial.
Second, the lower bound p (1/2)+ε of n in our corollary is very rough. How to improve the constant (1/2) is an interesting open problem. Conjecture 1. Let 0 < δ < 1 be a fixed positive number and p be a prime large enough.
en, for any positive integer p δ < n < p, there must exist two primitive roots α and β modulo p such that the equation α + β � n holds.

Several Lemmas
In order to complete the proof of the main result, we need several simple lemmas. For the sake of simplicity, we do not repeat some elementary number theory and analytic number theory results, which can be found in references [12][13][14]. First, we have the following.

Lemma 1.
Let p be an odd prime. en, for any integer a with (a, p) � 1, we have the identity Proof. In fact, this result is Lemma 17 in [15]. Some related works can also be found in [16][17][18][19].

Proof of the Theorem
Now, we shall complete the proof of our main result. For any integer 1 < n ≤ p − 1, from the definition of S(n, p) and Lemma 1, we have It is clear that χ t,k (a) � e(t · ind(a)/k) is a Dirichlet character modulo p. So, from the Polya and Vinogradov's classical work (see [12]; eorem 8.21 and eorem 13.15), we have the estimate where χ is any nonprincipal character modulo p. Now, from (11), (12), and Lemma 3, we have Journal of Mathematics 3 where we have used the identity and ω(m) denotes the number of all distinct prime divisors of m. is completes the proof of our theorem.

Conclusion
e main result in this paper is a theorem, which is closely related to Golomb's conjecture. It describes that when prime p is large enough, for any integer p (1/2)+ε ≤ n ≤ p − 1, there must exist two primitive roots α and β modulo p such that the equation n � α + β holds, where 0 < ε < (1/2) be a fixed positive number. At the same time, we also give a sharp asymptotic formula for the counting function of all such solutions (α, β). In fact, our conclusion is much stronger than Golomb's conjecture in the reduced residue system 1, 2, 3, . . . , p − 1 modulo p. As a note of the corollary, we also proposed an open problem.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.