On distribution properties of cubic residues

1.   Introduction

Let $p$ be an odd prime, and $a$ be an integer with $(a, p) = 1$. In order to study of the properties of quadratic residues modulo $p$, Legendre first introduced the characteristic function of the quadratic residues $\left(\frac{a}{p}\right)$, which was later called Legendre's symbol as follows:

The introduction of this symbol greatly promotes the research about the properties of the quadratic residues and non-residues modulo $p$. Therefore, a great number of scholars began to study related work and obtained a series of valuable research results. For example, if $p$ be an odd prime with $p = 4k+1$, then for any quadratic residue $r$ and quadratic non-residue $s$ modulo $p$, one has the identity (see ^[2]: Theorem 4–11)

where $a^{-1}$ is the inverse of $a\bmod p$. In detail, $a^{-1}$ satisfy the equation $x\cdot a\equiv 1\bmod p$.

Many other papers related to power residues and non-residues modulo $p$ can also be found in references ^{[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20]}, here we will not describe it in detail.

Recently, Wang Tingting and Lv Xingxing ^[5] studied the distribution properties of a certain quadratic residues and non-residues modulo $p$, and proved the following two conclusions:

Theorem A. For any prime $p$ with $p\equiv 3\bmod 4$, one has the identities

and

where $N(p, 1)$ denotes the number of all integers $1\leq a\leq p-1$ such that $a$, $a+a^{-1}$ and $a-a^{-1}$ are all quadratic residues modulo $p$, $N(p, -1)$ denotes the number of all integers $1\leq a\leq p-1$ such that $a$ is a quadratic non-residue modulo $p$, $a+a^{-1}$ and $a-a^{-1}$ are quadratic residues modulo $p$.

Theorem B. For any prime $p$ with $p\equiv 1\bmod 4$, one has the asymptotic formulas

and

where we have the estimates $|E(p, 1)|\leq \frac{3}{4}\cdot \sqrt{p}$, $|E_1(p, 1)|\leq \frac{5}{4}\cdot \sqrt{p}$, $|E(p, -1)|\leq \frac{3}{4}\cdot \sqrt{p}$ and $|E_1(p, -1)|\leq \frac{5}{4}\cdot \sqrt{p}$.

As two corollaries of these results, Wang Tingting and Lv Xingxing ^[5] solved two open problems proposed by professor Sun Zhiwei. That is, they proved:

For any prime $p\geq 101$, there is at least one integer $a$, such that $a$, $a+a^{-1}$ and $a-a^{-1}$ are all quadratic residues modulo $p$.

For any prime $p\geq 18$, there is at least one quadratic non-residue $a\bmod p$, such that $a+a^{-1}$ and $a-a^{-1}$ are quadratic residues modulo $p$.

In the field of the quadratic residues research, it is worth mentioning Z. H. Sun's important work ^[6]. We believe that Theorem A and B can also be derived from earlier results in ^[6]. All research works are very meaningful. In fact, the distribution of power residues plays an vital role in mathematics and cryptography, a number of important number theory and information security problems are closely related to it. A survey on this can be found in F. L. Ţiplea, S. Iftene, G. Teşeleanu, A. M. Nica ^[10]. Therefore, it is necessary to continue to study the distribution properties of the power residues.

Inspired by the works in ^[5] and ^[6], we will naturally ask that what happens to the cubic residues modulo $p$?

It is clear that if $(p-1, 3) = 1$, then for any integer $a$ with $(a, p) = 1$, the congruence equation $x^3\equiv a\bmod p$ has one solution. So it is a trivial situation in the condition of $(p-1, 3) = 1$. But if $p\equiv 1\bmod 3$, then what happens?

This paper focuses on such problems. Let $p$ be a prime with $p\equiv 1\bmod 3$, and $N(p)$ denotes the number of all integers $1 < a < p-1$ such that $a+a^{-1}$ and $a-a^{-1}$ are cubic residues modulo $p$. In this paper, using the elementary methods, the properties of the third-order characters and Gauss sums, and the estimate for character sums, we are going to give an exact identity and some asymptotic formulas for $N(p)$. Our main results are summarized as following three conclusions:

Theorem 1. Let $p$ be an odd prime with $p\equiv 7 \bmod 12$. If $2$ is a cubic residue $\bmod\; p$, then we have the identity

where $d$ is uniquely determined by $4p = d^2+27b^2$ and $d\equiv 1\bmod 3$.

Theorem 2. Let $p$ be an odd prime with $p\equiv 7 \bmod 12$. If $2$ is not a cubic residue $\bmod\; p$, then we have the asymptotic formula

where we have the estimates $|E(p)|\leq \frac{2}{3}\cdot \sqrt{p}$.

Theorem 3. Let $p$ be an odd prime with $p\equiv 1 \bmod 12$, then we have the asymptotic formula

where we have the estimates $|E_1(p)| < \frac{26}{9}\cdot \sqrt{p}$.

According to our theorems, we can deduce the following:

Corollary. Let $p > 700$ be a prime with $p\equiv 1\bmod 3$, then there exists at least one integer $1 < a < p-1$ such that $a+a^{-1}$ and $a-a^{-1}$ are cubic residues $\bmod\; p$.

2.   Several lemmas

To prove our main results, we need following several basic lemmas. For simplicity, there is no need to repeat some elementary knowledge of number theory and analytic number theory, which can be found in references ^[1,2,3].

Lemma 1. Let $p$ be a prime with $p\equiv 1 \bmod 3$. Then for any third-order character $\lambda\bmod p$, we have the identity

where $\tau(\chi) = \sum\limits_{a = 1}^{p-1}\chi(a)\ e\left(\frac{a}{p}\right)$ denotes the classical Gauss sums, and $e(y) = e^{2\pi iy}$, $d$ is uniquely determined by $4p = d^2+27b^2$ and $d\equiv 1\bmod 3$.

Proof. This result is Theorem 1 in Zhang Wenpeng and Hu Jiayuan ^[21].

Lemma 2. Let $p$ be an odd prime with $p\equiv 7 \bmod 12$. Then for any third-order character $\lambda\bmod p$, we have the identity

Proof. Note that $p\equiv3\bmod 4$ and $\left(\frac{-1}{p}\right) = -1$, so based on the properties of the Legendre's symbol and complete residue system $\bmod\; p$, we obtain

Similarly, we can also deduce that

Combining (2.1) and (2.2) we have the identity

This proves Lemma 2.

Lemma 3. Let $p$ be an odd prime with $p\equiv 1\bmod 3$. Then for any third-order character $\lambda\bmod p$, we have the identity

Proof. Write $\chi_2 = \left(\frac{*}{p}\right)$. Note that the identities $\overline{\lambda} = \lambda^2$, from the properties of the Gauss sums and the Legendre's symbol $\bmod\; p$ we have

Now according to the properties of complete residue system $\bmod\; p$, we have

On the other hand, from the properties of Legendre's symbol $\bmod\; p$ we also have

Since $\lambda^3(2) = 1$, so applying (2.4) and (2.5) we can deduce the identity

Note that $\tau^2(\chi_2) = \chi_2(-1)p$ and $\tau(\lambda)\tau\left(\overline{\lambda}\right) = \tau(\lambda)\overline{\tau(\lambda)} = p$, combining Lemma 1, (2.3) and (2.6) we can deduce the identity

This proves Lemma 3.

Lemma 4. Let $p$ be an odd prime with $p\equiv 1 \bmod 3$. Then for any third-order character $\lambda\bmod p$, we have the identity

Proof. It can be deduced from the same methods of proving Lemma 3.

Lemma 5. Let $p$ be an odd prime with $p\equiv 7 \bmod 12$. Then for any third-order character $\lambda\bmod p$, we have the identity

Proof. Note that the identity

From the properties of the Legendre's symbol $\bmod\; p$ we have

Applying (2.7) and Lemma 3 we may immediately get

This proves Lemma 5.

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 $p$ with $p\equiv 7\bmod 12$, note that $\left(\frac{-1}{p}\right) = -1$, so for all integers $1 < a < p-1$, we have $(a+a^{-1}, p) = 1$. From Lemma 2–5 we have

If $2$ is a cubic residue $\bmod\; p$, then $\lambda(2) = 1$, from (3.1) and Lemma 1 we have

This proves Theorem 1.

If $2$ is not a cubic residue $\bmod\; p$, then $1+\lambda(2)+\overline{\lambda}(2) = 0$, note that the estimate $|\tau(\lambda)| = \sqrt{p}$, from (3.1) and Lemma 1 we have

where $|E(p)| = \frac{1}{9p}\cdot \left| \left(2-\overline{\lambda}(2)\right)\tau^3(\lambda)+(2-\lambda(2))\cdot \tau^3\left(\overline{\lambda}\right) \right| \leq \frac{2}{3}\cdot \sqrt{p}$.

This proves Theorem 2.

If $p\equiv 1\bmod 12$, then there exist integer $r$ such that $r^2 = (-r)^2\equiv -1\bmod\; p$ and $r+r^{-1}\equiv 0\bmod\; p$. For this integer $1 < r < p-1$, we also have $\lambda(r) = \overline{\lambda}^2(r) = \overline{\lambda}(r^2) = \overline{\lambda}(-1) = 1$, so we have

From the Weil's work ^[22] we have the estimates

If $\lambda(2) = 1$, then note that $|d|\leq 2\cdot \sqrt{p}$, applying (3.2–3.4) and Lemma 1–5 we have asymptotic formula

where $|E_1(p)|\leq \frac{26}{9}\cdot \sqrt{p}$.

If $\lambda(2)\neq1$, then applying (3.2–3.4) and Lemma 2–5 we have asymptotic formula

where $|E_1(p)|\leq \frac{26}{9}\cdot \sqrt{p}$.

Now Theorem 3 follows from asymptotic formulas (3.5) and (3.6).

This completes the proofs of our all results.

4.   Conclusion

The main results of this paper are three theorems and a corollary. Theorem 1 obtained an exact formula for $N(p)$ with $p = 12k+7$ and $2$ is a cubic residue modulo $p$. Theorem 2 and Theorem 3 established two asymptotic formulas for $N(p)$ with $p\equiv 1\bmod 3$ and $2$ is not a cubic residue modulo $p$. At the same time, we also give two sharp upper bound estimates for the error terms. As some applications, we also deduced following corollary:

If $p > 700$ is a prime with $p\equiv 1\bmod 3$, then there is at least one integer $1 < a < p-1$ such that $a+a^{-1}$ and $a-a^{-1}$ are cubic residues modulo $p$.

Acknowledgments

The authors would like to thank the referee for their very helpful and detailed comments.

This work is supported by the N. S. F. (2017MS0114) of Inner Mongolia, Talent introduction research Foundation of Hetao College (HYRC2019007) 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.