The primitive roots and a problem related to the Golomb conjecture

1.   Introduction

Let $p$ be an odd prime, ${\bf{A}}(p)$ denotes the set of all primitive roots $g$ modulo $p$ with $1\leq g\leq p-1$. The Golomb conjecture (see ^[1]) in a reduced residue system modulo $p$ is whether there exist two primitive roots $\alpha$ and $\beta \in {\bf{A}}(p)$ such that the congruence

This conjecture has been basically solved in the finite field $\mathbb{F}_q$. Interested readers can refer to the references ^{[2,3,4,5,6,7,8,9,10,11,12,13]}. In fact, people have proved versions of the above result. Here, we simply describe one of them as follows: Let $p$ be an odd prime large enough. Then for any integers $a$, $b$ and $c$ with $abc$ coprime to $p$ (i.e., $(abc, p) = 1$), there are at least two primitive roots $\alpha$ and $\beta \; \bmod\; p$ such that the congruence $a\alpha+b\beta\equiv c\bmod p$ holds. See Qi Sun ^[3].

In this paper, we continue to work on this problem, because we find that the Golomb conjecture can be further strengthened. To make our problem more general, we will describe it in a finite field. Let $\mathbb{F}_q$ be a finite field of $q$ elements with characteristic $p$. Our problem in $\mathbb{F}_q$ can be summarized as follows:

For any two non-zero elements $a\neq b \in \mathbb{F}_q$, do there exist three primitive elements $\alpha$, $\beta$ and $\gamma \in \mathbb{F}_q$ such that the equations $\alpha+\gamma = a$ and $\beta+\gamma = b$ hold?

Obviously, if this problem is correct, then the Golomb conjecture must be true. The converse is not necessarily true. So, our problem can be seen as a further generalization and extension of the Golomb conjecture.

In this paper, we use elementary methods, properties of Gauss sums and estimates for character sums to give an affirmative answer to the above problem. To better describe our results, we use the counting function $N(\alpha, \beta; p)$, which denotes the number of all primitive roots $u$, $v$ and $w\in {\bf{A}}(p)$ such that the equations $u+w\equiv \alpha\bmod p$ and $v+w\equiv \beta\bmod p$ hold. Then we have the following result:

Theorem. Let $p$ be an odd prime. Then for any integers $1\leq \alpha\neq\beta\leq p-1$, we have the asymptotic formula

where as usual, $\phi(n)$ denotes the Euler function, and $\omega(n)$ denotes the number of all distinct prime divisors of $n$.

Obviously, our conclusion can also be generalized to the finite field $\mathbb{F}_q$. From our theorem we may immediately deduce the following:

Corollary. Let $p$ be an odd prime large enough. Then for any integers $1\leq a\neq b\leq p-1$, there exist three primitive roots $\alpha, \ \beta$ and $\gamma\bmod p$ such that the congruence equations

Obviously one can ask: Can the Golomb conjecture be extended further? Specifically, for three pairwise distinct nonzero elements $\alpha$, $\beta$ and $\gamma \in \mathbb{F}_q$, do there exist four primitive elements $a$, $b$, $c$ and $d\in \mathbb{F}_q$ such that the equations

We leave this as an open problem.

2.   Several lemmas

To complete the proof of our main result, we need following three 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 ^[14] and ^[15]. First, we have the following:

Lemma 1. Let $p$ be an odd prime. Then for any integer $a$ with $(a, p) = 1$, we have the identity

where $e(y) = e^{2\pi iy}$, $\mathop{{\sum_{r = 1}^{k}}\ '}$ denotes the summation over all integers $1\leq r\leq k$ such that $r$ is coprime to $k$, $\mu(n)$ is the Möbius function, and ${\rm{ind}}(a)$ denotes the index of $a$ relative to some fixed primitive root $g\bmod p$.

Proof. See Proposition 2.2 in ^[16].

Lemma 2. Let $p$ be a prime. Then for any integer $h$ with $(h, p) = 1$, we have the identity

where $\chi$ denotes a Dirichlet character modulo $p$, and $\tau(\chi) = \sum\limits_{a = 1}^{p-1}\chi(a) e\left(\frac{a}{p}\right)$ denotes the classical Gauss sums corresponding to $\chi$.

Proof. For integers $1\leq r\leq k\leq p-1$ with $k\mid p-1$ and $(r, k) = 1$, we write $e\left(\frac{r\cdot {\rm{ind}}(a)}{k}\right) = \chi_{r, k}(a)$, and $\chi_{r, k}(a) = 0$, if $p\mid a$. It is clear that $\chi_{r, k}(a)$ is a Dirichlet character modulo $p$. Note that by the properties of the classical Gauss sums we have

Applying the above with $\chi: = \chi_{r, k}$ and using Lemma 1, we immediately deduce the identity

This proves Lemma 2.

Lemma 3. Let $p$ be an odd prime, $\chi_1, ..., \ \chi_r$ be Dirichlet characters modulo $p$, at least one of which is non-principal character. Let $f(x)$ be an integral coefficient polynomial of degree $d$. Then for pairwise distinct integers $a_1, ..., a_r$, we have the estimate

Proof. This is Lemma 17 in ^[17]. Some related work can also be found in ^[18].

Lemma 4. Let $p$ be a prime. Then for any integer $d$ with $(d, p) = 1$, we have the estimate

Proof. Note that $|\tau(\chi)| = \sqrt{p}$, if $\chi$ is any non-principal character modulo $p$, and $|\tau(\chi)| = 1$, if $\chi$ is the principal character modulo $p$. From the identity

and Lemma 2, we have

This proves Lemma 4.

Lemma 5. Let $p$ be a prime. Then for any integers $1\leq \alpha \neq\beta\leq p-1$, we have the estimate

Proof. Note that $|\tau(\chi)| = 1$, if $\chi$ is the principal character modulo $p$. And if $(v, p) = 1$, $u$ pass through a reduced residue system modulo $p$, then $uv$ also pass through a reduced residue system modulo $p$. So from Lemma 2, Lemma 3 and the methods of proving Lemma 4 we have

This proves Lemma 5.

3.   Proof of the theorem

In this section, we shall complete the proof of our main result. For any integers $1\leq \alpha \neq\beta\leq p-1$, note that the trigonometric identity

and

Thus, from Lemma 4 and Lemma 5 we have

This completes the proof of our theorem.

4.   Conclusion

The main result of this paper is a theorem, which is closely related to the Golomb conjecture. It describes that when the prime $p$ is large enough, for any integers $1\leq \alpha\neq \beta\leq p-1$, there exist three primitive roots $u, \ v$ and $w \in {\bf{A}}(p)$ such that the congruence equations $u+w \equiv \alpha\bmod p\ {\rm{ and }} \ v+w \equiv \beta\bmod p$ hold. At the same time, we also give a sharp asymptotic formula for the counting function of all such solutions $(u, v, w)$. Of course, our conclusion can also be generalized to the finite field $\mathbb{F}_q$. In order to further study the content related to the Golomb conjecture, we also proposed an open problem.

Acknowledgments

The authors would like to thank the referee for their very helpful and detailed comments. In particular, many English grammar and error correction, so that the text reads more smoothly.

This work is supported by the Y. S. T. N. S. P (2019KJXX-076), the N. S. B. R. P. (2019JM-207) of Shaanxi Province and the N. S. F. (11771351) of P. R. China.

Conflict of interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.