On a Certain Quadratic Character Sums of Ternary Symmetry Polynomials

Abstract

In this article, we are using the elementary methods and the properties of the classical Gauss sums to study the calculating problem of a certain quadratic character sums of a ternary symmetry polynomials modulo and obtain some interesting identities for them.

1. Introduction

Let be an odd prime, denotes the Legendre symbol , i.e., for any integer , one has

Some of the most commonly used properties of the Legendre symbol are as follows (see [1, 2]):where and are two different odd primes.

The introduction of the Legendre symbol has not only enriched the content of number theory but also greatly promoted the development of elementary and analytic number theory, especially the research on the properties of primes. For example, if is a prime with , then for any integers and with , one has the identity (see Theorems 4–11 in [2])

From (3), we naturally wonder, for other forms of primes , can they also be expressed in terms of Legendre’s symbol?

In particular, if is an odd prime with , then there are two integers and such that the identity (see [3])where is uniquely determined by .

In addition, if is an odd with , then there are two integers and such that

Although we have not found the representations of and or and in terms of the Legendre symbol modulo , we found that a certain quadratic character sum of the ternary symmetry polynomials are closely related to the numbers and .

In this paper, we shall use elementary methods and the properties of the classical Gauss sums to study the calculating problem of a certain quadratic character sum of binary symmetry polynomials modulo and obtain several interesting identities for them. That is, we shall prove the following results.

Theorem 1. Let be an odd prime with . Then, we have

Theorem 2. Let be an odd prime with . If 2 is a cubic residue modulo , then we have the identitywhere is the same as defined in (4).

Theorem 3. Let be an odd prime with . If 2 is not a cubic residue modulo , then we have the identityFrom these theorems, we may immediately deduce the following three corollaries.

Corollary 1. Let be an odd prime with . Then, we have

Corollary 2. Let be a prime with . If 2 is a cubic residue modulo , then we have

Corollary 3. If is an odd prime with , if 2 is not a cubic residue modulo , then we haveNotes: it is clear that our method is applicable to multivariate symmetry polynomials . But, when is larger, the calculation is more complicated, so we do not give it.

Theorem 3 is flawed. In other words, it gives us two possibilities. We do not know for sure which one is the exact value. How to determine its exact value is an open problem. Interested readers are encouraged to join us in the research.

2. Several Lemmas

In this section, we first give several simple lemmas. Of course, the proofs of these lemmas need some knowledge of elementary and analytic number theory. They can be found in many number theory books, such as [1, 2]. Other papers related to Gauss sums and character sums can also be found in [4–11]; here, we do not need to list.

Lemma 1. Let be a prime with . Then, for any third-order character , one has the identitywhere is the same as defined in (4).

Proof. See the work of Zhang and Hu [12].

Lemma 2. Let be a prime with . Then, for any sixth-order character , one has the identitywhere and is the same as defined in (4).

Proof. This result is Lemma 3 in the work of Chen [4], so we omit the proof process.

Lemma 3. Let be a prime with . Then for any third-order character , we have the identitywhere denotes the Legendre symbol modulo .

Proof. Note that and ; from the properties of Gauss sums and the Legendre symbol modulo , we haveIt is clear that andand from the properties of the Gauss sums, we can getOn the other hand, note that ; we also haveNote that ; from (17)–(19), we have the identitySimilarly, we also getNow, combining (15), (20), and (21), we have the identityThis proves Lemma 3.

Lemma 4. Let be a prime with . Then, for any third-order character , we have the identity

Proof. Note that ; from the methods of proving Lemma 3, we haveApplying (18) and (19), we haveSimilarly, we also haveCombining (24)–(26), we haveTaking the conjugate in (27), we haveNow, Lemma 4 follows from (27) and (28).

Lemma 5. Let be a prime with . Then, for any third-order character , we have the identities

Proof. Note that ; from (18), (19), and the methods of proving Lemma 3, we haveCombining (30)–(32), we haveTaking the conjugate given above, we can deduce the other identity. This proves Lemma 5.

Lemma 6. Let be a prime with . Then, for any third-order character , we have the identity

Proof. From (18), (19), and the methods of proving Lemma 3, we haveCombining (35)–(37), we haveThis proves Lemma 6.

3. Proofs of the Theorems

Now, we prove our theorems. From the properties of the reduced residue system modulo , we have

If , when passes through a reduced residue system modulo , then also passes through a reduced residue system modulo . So, from (39) and (15), we have the identity

This proves Theorem 1.

If , let be a third-order character modulo ; then, for any integer with , we have the identity

So, from the symmetry of and , (39), and Lemmas 3–6, we have

If 2 is a cubic residue modulo , then . Note that is a sixth-order character modulo , , and , if ; and , if . From (42) and Lemmas 1 and 2, we have

This proves Theorem 2.

Now, we prove Theorem 3. If 2 is not a cubic residue modulo , then we have or . From (42) and Lemmas 1 and 2, we have

Since or , we haveor

So, from (45)–(47) and Lemma 1, we have

Combining (44) and (48), we have the identity

This completes the proof of Theorem 3.

4. Conclusions

The main results of this paper are three identities involving a certain quadratic character sums of ternary symmetry polynomials modulo . Theorem 1 proved an exact identity for the sum, if . Theorem 2 discussed the case and obtained an exact identity, provided 2 is a cubic residue modulo . Theorem 3 also discussed the case , where 2 is not a cubic nonresidue modulo . In this case, two possibilities are given. These results not only give the exact values of a certain quadratic character sums of ternary polynomials modulo but also some new contribution to research in related fields.

Data Availability

No data were used in this paper.

Conflicts of Interest

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

Acknowledgments

This work was supported by the N. S. F. (11771351) of P. R. China.