On the Sixth Residues and Some New Properties of Their Distribution

Abstract

In this paper, we use the analytic methods, the properties of the sixth-order characters, and the classical Gauss sums to study the computational problems of a certain special sixth residues’ modulo and give two exact calculating formulas for them.

1. Introduction

Let be an odd prime and be a fixed positive integer. For any integer a with , if the congruence equation has a solution , then we call a is a th residue modulo . Otherwise, a is called a th nonresidue modulo . In particular, if , 3, and 4, we call is a quadratic residue, cubic residue, and quartic residue modulo , respectively. Undoubtedly, the research of quadratic residue is the most concerned topic. Legendre first introduced the characteristic function of the quadratic residues modulo , which later was called Legendre’s symbol. It is defined as follows:

Sometimes, we write Legendre’s symbol as for the sake of writing. This is because the introduction of this function greatly facilitated the study of quadratic residue properties and promoted the development of elementary number theory and analytic number theory. This is especially true in the study of primes and related problems. For example, if is a prime with , then one has (see Theorem 4–11 in [1])where denotes the inverse of a. That is, , and .

Of course, there are many papers involving quadratic residues and primes, so we cannot cover all of them. Those who are interested can refer to [2–9].

In this paper, we are concerned with the problem of whether the special integers and both are th residues’ modulo . Let denote the number of all integers such that and both are th residues’ modulo . Then, how are the values of distributed?

Very recently, some authors had studied the calculating problem of and obtained a series of interesting results. For example, Wang and Lv [10] obtained the identity

Hu and Chen [11] proved the following result: let be an odd prime with . If 2 is a cubic residue , then one has the identity

If 2 is not a cubic residue , then one has the asymptotic formulawhere is defined in (7) and satisfies the estimates .

Su and Zhang [12] considered the case and proved the identity

As an extension of [10–12], a natural problem is what about sixth residues modulo ? It is clear that if , then the problem is trivial. That is, any quadratic residue a modulo is a sixth residue modulo . So, we just consider the nontrivial case . In this case, we know that there are two integers and such that the identitywhere is uniquely determined by (see [13]).

And, it is clear from (7) that the value of must be related to and .

In this paper, we will use the analytic methods, the properties of the classical Gauss sums, and the estimate for some special character sums to study the computational problems of and give an exact calculating formula for it. That is, we will prove the following two results.

Theorem 1. Let be an odd prime with . If 2 is a cubic residue modulo , then we have the identitywhere is the same as defined in (7).

Theorem 2. Let be an odd prime with . If 2 is not a cubic residue modulo , then we haveFrom our theorems, we may immediately deduce the following two corollaries.

Corollary 1. Let be an odd prime with . If 2 is a cubic residue modulo , then we have the congruence

Corollary 2. Let be an odd prime with . If 2 is not a cubic residue modulo , then we have the congruence

First, in Theorems 1 and 2, we must distinguish whether 3 is a cubic residue modulo because of the need to calculate the character sums. In different cases, the values of character sums are different.

Second, if is a prime with , then, for some character sums, we can only use Weil’s classical work [14, 15] to get some upper bound estimates and we cannot get their exact values. So, in this case, we can only deduce a sharp asymptotic formula for . That is,

Third, if is an odd prime with and 2 is not a cubic residue modulo , then our Theorem 2 also obtained an exact calculating formula for , which is obviously better than the corresponding result in [11].

Of course, our Theorem 2 is flawed, and it presents two possibilities. How to determine its correct value is an interesting open problem.

Finally, if is a prime with , then we know that 2 is a cubic residue modulo if and only if . That is, is an even number. Otherwise, is an odd number. Especially for primes , after some simple calculations, we have , , , , , , and . Since is an integer, so applying Corollary 2, we can get the congruences: , , , , , , and .

Now, we consider Legendre’s symbol . Note that , , , , , , and . From the above congruences and these values, we have a reason to believe the following.

Conjecture. Let be an odd prime with . If 2 is not a cubic residue modulo , then we have the identity

2. Several Lemmas

In this section, we decompose the proofs of our theorems into the following several lemmas. For the sake of simplicity, the basic knowledge required in this section is not listed, and only three necessary references [1, 16, 17] are provided here. First, we have the following.

Lemma 1. Let be an odd prime with . Then, for any third-order character modulo (i.e., and , the principal character modulo ), we have the identitywhere , is uniquely determined by , denotes the classical Gauss sums, and .

Proof. For the proof of this lemma, see Zhang and Hu [18] or Berndt and Evans [19].

Lemma 2. Let p be a prime with . Then, for any third-order character and sixth-order character (i.e., , , and ), we have the identity

Proof. From the properties of the Gauss sums and the reduced residue system modulo , note that the identityand we haveOn the contrary, we also haveNote that identity , and from (17) and (18), we deduce the identityThis proves Lemma 2.

Lemma 3. Let be an odd prime with . Then, for any third-order character , we have the identity

Proof. Note that , , and , and from Lemma 2, properties of the Gauss sums, and Legendre’s symbol , we havewhere we have used the identity .
Similarly, we can also deduce thatIt is clear that Lemma 3 follows from (21) and (22).

Lemma 4. Let be an odd prime with . Then, we have the identity

Proof. From the methods of proving Lemma 2 and the properties of the Gauss sums, we haveSo, from (24), we haveThis proves Lemma 4.

Lemma 5. Let be an odd prime with . Then, we have

Proof. It is the same as the proof of Lemma 4, so it is omitted.

Lemma 6. Let be an odd prime with . Then, we have the identities

Proof. Note that , and from the properties of the reduced residue system modulo , we havewhich implies thatThis proves Lemma 6.

Lemma 7. Let be an odd prime with . For any sixth-order character , we have

Proof. Since , , so, from the reduced residue system modulo , we havewhich implies thatSimilarly, we can also deduce the identityThis proves Lemma 7.

Lemma 8. Let be an odd prime with . Then, for any third-order character and , we have

Proof. Note that and , we haveSo, we have the identitySimilarly, we can also deduce the identityThis proves Lemma 8.

Lemma 9. Let be an odd prime with . Then, for any third-order character , we have

Proof. From the properties of Legendre’s symbol , we haveSimilarly, we also haveCombining (39) and (40), we have the identityThis proves Lemma 9.

3. Proofs of the Theorems

In this section, we shall complete the proofs of our main results. First, we prove Theorem 1. For any prime with , let denote a third-order character modulo ; then, is a sixth-order character modulo . So, for any integer a with , from the characteristic functionand note that , , , , , and , and we have the identity

Note that and , so we haveFrom Lemma 1, 4, and 5, we have

Now, if 2 is a cubic residue modulo , then . Combining (43)–(46), Lemma 3, and Lemma 6–9, we have

This proves Theorem 1.

Now, we prove Theorem 2. If 2 is not a cubic residue modulo , then . That is to say,where .

In this case, from Lemma 1, we haveor

From (48), (50), and Lemma 3, we have

Combining (43)–(46), (51), and Lemma 6–9, we haveor

This completes the proofs of our all results.

4. Conclusion

The main results of this paper are two theorems and two corollaries. Theorem 1 gives an exact computing formula for with and 2 is a cubic residue modulo . If and 2 is not a cubic residue modulo , then Theorem 2 established an identity for and there are two possibilities. As some applications of these theorems, we also deduced two interesting congruences. For example, one of them iswhere and 2 is a cubic residue modulo .

In addition, if and 2 is not a cubic residue modulo , then we also have an interesting conjecture. That is,

Data Availability

No data were used to support the findings of the study.

Conflicts of Interest

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

Authors’ Contributions

All authors contributed equally to this work. All authors read and approved the final manuscript.

Acknowledgments

This work was supported by Natural Science Project ZK20-89, Yangling Vocational and Technical College, and “the Xi’an Science and Technology Plan Innovation Fund” Arts and Sciences Special Project 2020KJWL08.