Abstract
In this paper, we use the elementary methods and the estimates for character sums to prove the following conclusion. Let be a prime large enough. Then, for any positive integer with , there must exist two primitive roots and modulo with such that the equation holds, where is a fixed positive number. In other words, can be expressed as the exact sum of two primitive roots modulo .
1. Introduction
Let be a prime and be a finite field of () elements with characteristic . The Golomb conjecture (see [1]) can be summarized as follows: for any nonzero element , there exist two primitive elements and such that the equation holds.
If we assume that denotes the set of all primitive roots modulo with , then the Golomb conjecture in a reduced residue system modulo can be described as that, for any integer , there exist two primitive roots and such that the congruence holds.
This conjecture is not only basically solved but also carried on various generalizations. Interested readers can refer to the references [2–11]. For example, let be an odd prime large enough. Then, for any integers , , and with , there are at least two primitive roots and such that the congruence holds (see Sun [2]).
It is clear that if integer and the primitive roots satisfy the congruence , then or .
A natural question is whether for a fixed , there are two primitive roots of such that
Of course, for some positive integers , equation (1) has no solutions. For example, , and 3. So, we think that the problem in (1) is meaningful, and it is also closely related to the minimum primitive root modulo .
On the other hand, we also want to know how large is (relative to ), so that equation (1) must have a solution.
For the sake of convenience, for any odd prime and integer , let denote the number of all solutions of the equation , where and are two primitive roots modulo with .
In this paper, we shall use the elementary methods and the estimates for character sums to study the asymptotic properties of and prove the following.
Theorem 1. Let be an odd prime. Then, for any integer , we have the asymptotic formula:where, as usual, denotes the Euler function and denotes the number of all distinct prime divisors of .
It is clear that, for any positive number , if prime is large enough, then our theorem is nontrivial for all integers . That 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 be an odd prime large enough, be a fixed positive number. Then, for any positive integer with , there must exist two primitive roots and modulo such that
Note: first, the conclusion in our theorem can also be generalized. That is, let be an odd prime and be a fixed positive integer. For any integers , if denotes the number of all solutions of the equations , , where and all () are the primitive roots modulo , then we have the following asymptotic formula:
If and , then this asymptotic formula is nontrivial.
Second, the lower bound of in our corollary is very rough. How to improve the constant is an interesting open problem.
Conjecture 1. Let be a fixed positive number and be a prime large enough. Then, for any positive integer , there must exist two primitive roots and modulo such that the equation holds.
2. 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 [12–14]. First, we have the following.
Lemma 1. Let be an odd prime. Then, for any integer with , we have the identitywhere , denotes the summation over all integers such that is coprime to , is the Möbius function, and denotes the index of relative to some fixed primitive root .
Proof. See Proposition 2.2 in [13].
Lemma 2. Let be an odd prime and be Dirichlet characters modulo , at least one of which is nonprincipal character. Let be an integral coefficient polynomial of degree . Then, for pairwise distinct integers , we have the estimate
Proof. In fact, this result is Lemma 17 in [15]. Some related works can also be found in [16–19].
Lemma 3. Let be an odd prime. Then, for any integer and any two Dirichlet characters and (at least one of which is nonprincipal character) modulo , we have the estimate
Proof. It is clear that, for any integer , we have the trigonometric identityand the estimateFrom (8), (9), and Lemma 2, we haveThis proves Lemma 3.
3. Proof of the Theorem
Now, we shall complete the proof of our main result. For any integer , from the definition of and Lemma 1, we have
It is clear that is a Dirichlet character modulo . So, from the Polya and Vinogradov’s classical work (see [12]; Theorem 8.21 and Theorem 13.15), we have the estimatewhere is any nonprincipal character modulo .
Now, from (11), (12), and Lemma 3, we havewhere we have used the identityand denotes the number of all distinct prime divisors of .
This completes the proof of our theorem.
4. Conclusion
The main result in this paper is a theorem, which is closely related to Golomb’s conjecture. It describes that when prime is large enough, for any integer , there must exist two primitive roots and modulo such that the equation holds, where be a fixed positive number. At the same time, we also give a sharp asymptotic formula for the counting function of all such solutions . In fact, our conclusion is much stronger than Golomb’s conjecture in the reduced residue system modulo . 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
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Authors’ Contributions
All authors have equally contributed to this work. All authors read and approved the final manuscript.
Acknowledgments
This work was supported by the NSF of P. R. China (no. 11771351) and the Natural Science Foundation of Hebei Province (no. A2015410006).