#### Abstract

The main purpose of this article is using the elementary methods and the properties of the character sums to study the calculating problem of the number of the solutions for one kind congruence equation modulo (an odd prime) and give some interesting identities and asymptotic formulas for it.

#### 1. Introduction

Let be a prime. For any integer , the Legendre’s symbol modulo is defined as follows:

This arithmetical function occupies a very important position in the elementary number theory and analytic number theory, and many classical number theory problems are closely related to it, for example, the least quadratic nonresidue problem, the class number formula of the quadratic field, and the prime number structure. In particular, if is a prime with , then we have the identity (see Theorems 4–11 in [1])where is any quadratic nonresidue modulo , and .

In addition to its wide range of applications, the Legendre’s symbol has a number of interesting and important properties of its own. The quadratic reciprocity law is one of them. That is, for any two odd primes and with , we have the identity (see Theorem 9.8 in [2])

Some other papers related to character sums can be referred to [3–9]. There is no need to give detailed examples of their contents here, the interested readers may refer to them.

In this paper, we will consider the following elementary number theory problem related to -th residue modulo , in particular, the quadratic residue modulo .

For any integer with , let , and denote the number of the solutions of the congruence equation () such that and .

Theorem 1. *Let be an odd prime. Then, for any integer with , we have the identity**For any fixed positive integer and any prime with , if we let denote the number of the solutions of the congruence equation with , both and are the -th residues modulo . Then, we also have the following.*

Theorem 2. *Let be a prime with . Then, for any integer with , we have the identityif is a cubic residue modulo , then we havewhere is any third-order character modulo , denotes the classical Gauss sums, , and is uniquely determined by .*

Theorem 3. *Let be a prime with . If integer is a quadratic residue modulo , then have the identitywhere is defined as the same as in (2).*

It is clear that from Theorem 1, we can deduce the following several corollaries.

Corollary 1. *Let be an odd prime. Then, for any integer with , we have the identity*

Corollary 2. *Let be an odd prime. Then, for any integer with , we have the identity*

Corollary 3. *Let be an odd prime. Then, for any integer with , we have the identity**Note that the estimate and , from Theorem 2 and Theorem 3; we can also deduce the following corollaries.*

Corollary 4. *Let be an odd prime with . Then, for any integer with , we have the asymptotic formula*

Corollary 5. *Let be a prime with . Then, for any cubic residue modulo , we have the identity*

Corollary 6. *Let be a prime with . Then, for any quadratic residue , we have the asymptotic formula**Some notes: it is very interesting that the right hand side of the identity in Corollary 3 does not depend on constant . Of course, the research content and methods in this paper can be further promoted.*

#### 2. Several Lemmas

In this section, we will give several necessary lemmas. Of course, the proofs of the theorems and these lemmas need the knowledge of elementary and analytic number theory, in particular, the properties of the quadratic residues modulo and the classical Gauss sums. All these can be found in references [1, 2, 10], and we do not repeat them. First, we have the following.

Lemma 1. *Let be an odd prime with . Then, for any three-order character , we have the identitywhere , and is uniquely determined by ,*

*Proof. *See the study by Zhang and Hu [4] or Berndt and Evans [5].

Lemma 2. *Let be an odd prime with . Then, we have**If is a cubic residue modulo , then we havewhere is defined as the same as in Lemma 1.*

*Proof. *For any integer , note that and ; from the properties of the three-order character and the classical Gauss sums modulo , we haveIf is a cubic residue modulo , then we have and ; from (17) and Lemma 1, we haveNow, Lemma 2 follows from (18).

Lemma 3. *Let be an odd prime with . Then, for any four-order character modulo , we have the identitywhere is defined as in (2).*

*Proof. *See Lemma 2.2 in the study by Chen and Zhang [7].

#### 3. Proofs of the Theorems

Applying three simple lemmas in Section 2, we can easily complete the proofs of our theorems. First, we prove Theorem 1. Note that the trigonometric identity, so from the definition of and the properties of the Legendre’s symbol, we havewhere is the Legendre’s symbol modulo . This proves Theorem 1.

Now, we prove Theorem 2. For any integer , from the properties of the third-order character modulo , we have

Applying formulas (20) and (22) and Lemma 2, the definition of , and the properties of the classical Gauss sums, we have

If is a cubic residue modulo , then we have . From (23) and Lemma 1, we may immediately deduce

Now, Theorem 2 follows from (23) and (24).

Similarly, using the methods of proving Theorem 2, we can also deduce Theorem 3. In fact, let be a four-order character modulo , then . From the properties of the four-order character, we have

Note that ; from (25) and the methods of proving Theorem 2, we have

If , is a quadratic residue modulo , then we have and . Thus, from (26) and Lemma 3, we have

If , is a quadratic residue modulo , then we have , , and . Thus, from (26) and Lemma 3, we have

Now, Theorem 3 follows from (27) and (28).

This completes the proofs of all our results.

#### Data Availability

The data used to support the findings of this study are included within the article.

#### 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 (11771351) of P. R. China.