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1.   Introduction

For a prime $p\equiv 1(\bmod 3)$, let $\mathbb{F}_p$ be the finite field of residues $(\mathrm{mod}p)$, let $G$ be the multiplicative group of non-zero residues $(\mathrm{mod}p)$ and let $H$ be the subgroup of non-zero cubic residues $(\mathrm{mod}p)$. For any $a\in G$, we defined the sums

and

where $\chi$ is a multiplicative character of order 3 over $\mathbb{F}_p$ and $e(x) = e^{2\pi\mathrm{i}x}$ in this paper. Both $S(a)$ and $G(\chi)$ are called Gauss sums of order 3. Gauss sums is very important in the analytic number theory and related research filed. Many scholars studied its properties and obtained a series of interesting results (see ^{[5,6,8,9,10,11,13]}).

Let $z\in G\setminus H$. By a classical result of Gauss ^[4] (also see Theorem 4.1.2 of ^[1]), $S(1), S(z)$ and $S(z^2)$ are three roots of the cubic equation

where $c$ is uniquely determined by

However, how to determine which of the three roots corresponds to $S(1)$ is still an open problem.

In this paper, for a fixed $z\in G\setminus H$, we find a relation between $S(1), S(z)$ and $S(z^2)$.

Theorem 1.1. Let $p\equiv 1(\bmod 3)$ and $z\in G\setminus H$. Then

where $\theta_p = \frac{1}{3}\arccos\left(-\frac{c}{2\sqrt{p}}\right)+j_p\frac{2}{3}\pi$; $j_p$ is one of three values $-1, 0, 1$ and only dependent on $p$; $c$ and $d$ are uniquely determined by

Moreover, there is a unique multiplicative character $\chi$ of order 3 over $\mathbb{F}_p$ such that

As application, we consider some congruence equations $\bmod p$. For $a_1, a_2, a_3\in G$, let $M(a_1, a_2, a_3)$ be the number of solutions of

and let $N(a_1, a_2, a_3)$ be the number of solutions of

In ^[2], Chowla, Cowles and Cowles showed that $M(1, 1, 1) = p^2+c(p-1)$. As pointed out in ^[3], the following is essentially included in the derivation of the cubic equation of periods by Gauss ^[4]: For a prime $p\equiv 1(\bmod 3)$ and for $z\in G\setminus H$, then one has

where $c$ and $d$ are uniquely determined by (1.1) (except for the sign of $d$).

Chowla, Cowles and Cowles ^[3] determined the sign of $d$ for the case $2\in G\setminus H$ as the following result shows.

Proposition 1.2. ^[3] Let a prime $p\equiv 1(\bmod 3)$. If $2\in G\setminus H$, then for any $z\in G\setminus H$, one has

where $c$ and $d$ are uniquely determined by (1.1) with

and

Recently, Hong and Zhu ^[7] solve the Gauss sign problem. In fact, they gave the following result.

Proposition 1.3. ^[7] Let a prime $p\equiv 1(\bmod 3)$ and $z\in G\setminus H$. Let $g$ be a generator of the multiplicative group $G$. one has

where $c$ and $d$ are uniquely determined by (1.1) with $d > 0$ and

Here $r_1$ and $r_2$ are uniquely determined by

Indeed, their result need to use the generator of group $G$ (that is the primitive root of module $p$). However, for a large prime $p$, it is not easy to find the primitive root of module $p$. In this paper, we consider $M(a_1, a_2, a_3)$, $N(a_1, a_2, a_3)$ and determine the sign of $d$ immediately by the coefficients $a_1, a_2$ and $a_3$. We have the following three more general results.

Theorem 1.4. Let a prime $p\equiv 1(\bmod 3)$ and $a_1, a_2, a_3\in G$.

(1) For the case $a_1a_2a_3\in H$, $M(a_1, a_2, a_3) = p^2+c(p-1)$;

(2) For the case $a_1a_2a_3\notin H$, $M(a_1, a_2, a_3) = p^2+\frac{1}{2}(p-1)(9d-c)$,

where $c$ and $d$ are uniquely determined by

Theorem 1.5. Let $p\equiv 1(\bmod 3)$ and $a_1, a_2, a_3\in G$.

(1) For the case $a_1a_2a_3\in H$,

(2) For the case $a_1a_2a_3\notin H$,

where $c$ and $d$ are uniquely determined by (1.3).

Corollary 1.6. Let $p\equiv 1(\bmod 3)$ and $a_1, a_2, a_3\in G$. Then

In ^[14], H. Zhang and W. P. Zhang proposed the following open problem:

Can the number of solutions to the cubic congruence equation

be calculated when $z\in G$?

Let $L(z)$ be the number of solutions of the above Eq (1.4). In ^[12], W. P. Zhang and J. Y. Hu proved that

However, in ^[12], they also proposed an interesting open problem: How to determine the choice of sign in (1.5). In this paper, we solve the sign problem in (1.5), and get the following result.

Theorem 1.7. Let $p$ be a prime number and $p\equiv 1(\bmod 3)$, let $z\in G\setminus H$. Then

where $c$ and $d$ are uniquely determined by

2.   Some useful lemmas

Lemma 2.1 (Theorem 3.1.3 of ^[1]). Let $p\equiv 1(\mathrm{mod} 3)$ and $\chi$ be a multiplicative character of order 3 over $\mathbb{F}_p$. Then

where the Jacobi sum $J(\chi, \chi) = \sum_{a = 1}^{p-1} \chi(a)\chi(1-a)$, $c$ and $d$ are uniquely determined by

with $g$ being the generator of the multiplicative group $G$ of non-zero residues $(\mathrm{mod}p)$ such that $\chi(g) = \frac{-1+\sqrt{3}\mathrm{i}}{2}$.

Lemma 2.2 (Lemma 4.1.1 of ^[1]). Let $p\equiv 1(\bmod 3)$. Let $g$ be a generator of the multiplicative group $G$ of non-zero residues $(\bmod p)$ with $\chi(g) = \frac{-1+\sqrt{3}\mathrm{i}}{2}$. Then

Lemma 2.3. Let $p\equiv 1(\mathrm{mod} 3)$ and $z\in G\setminus H$. Then there is a unique multiplicative character $\chi$ of order 3 over $\mathbb{F}_p$ such that

where $c$ and $d$ are uniquely determined by (1.2).

Proof. Let $g'$ be a generator of the group $G$. Note that $z\in G\setminus H$. So we have $\mathrm{ind}_{g'}z\equiv \pm 1 (\bmod 3)$. If $\mathrm{ind}_{g'}z\equiv 1 (\bmod 3)$, we take $g = g'$; If $\mathrm{ind}_{g'}z\equiv -1 (\bmod 3)$, we take $g = (g')^{-1}$. Hence $g$ also is a generator of the group $G$ and $\mathrm{ind}_gz\equiv 1 (\bmod 3)$. Thus we have

We take the multiplicative character $\chi(\cdot) = e\left(\frac{\mathrm{ind}_g(\cdot)}{3}\right)$. Obviously, we have

Obviously, all of the multiplicative non-principal characters of order 3 over $\mathbb{F}_p$ are $\chi$ and $\overline{\chi}$, $\overline{\chi}(z) = \overline{\chi(z)} = \frac{-1-\sqrt{3}\mathrm{i}}{2}$. Thus $\chi$ is the unique multiplicative character of order 3 over $\mathbb{F}_p$ with $\chi(z) = \frac{-1+\sqrt{3}\mathrm{i}}{2}$.

Note that $G^3(\chi) = pJ(\chi, \chi)$ by Lemma 2.2. Finally, using the Lemma 2.1, one immediately arrive the Lemma 2.3 as required.

Lemma 2.4. Let $\chi$ be a multiplicative character of order 3. Then for any $a\in G$, we have

Proof. Let $\chi$ be any multiplicative character of order 3. Then we have

Thus for any $a\in G$, we have

3.   Proof of Theorem 1.1

In this section, we prove Theorem 1.1. First, by Lemma 2.3, there is a unique multiplicative character $\chi$ of order 3 such that

where $c$ and $d$ are uniquely determined by (1.2). We can rewrite $G^3(\chi)$ by argument, and get

where $\theta = \frac{1}{3}\arccos(-\frac{c}{2\sqrt{p}}).$ Thus we have

where $j$ is one of three values $-1, 0, 1.$ Let $j_p = sgn(d)j$. Thus we have

Next, we will prove that $j_p$ does not depend on the sign of $d$. Note that $G(\overline{\chi}) = \chi(-1)\overline{G(\chi)} = \sqrt{p}e^{-\mathrm{i}sgn(d)(\theta+j_p\frac{2}{3}\pi)}$. By Lemma 2.4, we have

Obviously, by the definition of $S(1)$, the value of $S(1)$ doesn't depend on the sign of $d$. Thus we have that $j_p$ does not depend on the sign of $d$.

Take $\theta_p = \theta+j_p\frac{2}{3}\pi$. We have $G(\chi) = \sqrt{p}e^{\mathrm{i}sgn(d)\theta_p}$ and $S(1) = 2\sqrt{p}\cos(\theta_p)$. By Lemma 2.4, we have

Similarly, we have

This completes the proof of the Theorem 1.1.

4.   Some applications for cubic congruence equations and an example

In this section, we prove Theorem 1.4, 1.5 and 1.7. First, we begin with the proof of Theorem 1.4.

Proof of Theorem 1.4. By the orthogonality of additive character, we have

Then by Lemma 2.4, for any multiplicative character $\chi$ of order 3, we have

If $a_1a_2a_3\in H$, thus we have $\chi(a_1a_2a_3) = \overline{\chi}(a_1a_2a_3) = 1$. Then by Lemma 2.3, we have

If $a_1a_2a_3\in G\setminus H$, then by Lemma 2.3, we can take multiplicative character $\chi$ of order 3 satisfying

where $c$ and $d$ are uniquely determined by (1.3). Thus we have

This completes the proof of the Theorem 1.4.

Proof of Theorem 1.5. We have

If $a_1\equiv a_2(\bmod H)$, the number of solutions of the congruence equation $x^3\equiv a_1\overline{a}_2(\bmod p)$ is exactly 3. Thus we have

If $a_1\not\equiv a_2(\bmod H)$, the congruence equation $x^3\equiv a_1\overline{a}_2(\bmod p)$ has no solution. Thus we have

Hence Theorem 1.5 immediately follows from Theorem 1.4.

Proof of Theorem 1.7. First, by Lemma 2.3, there is a unique multiplicative character $\chi$ of order 3 such that

where $c$ and $d$ are uniquely determined by (1.2).

Note that $\chi(-1) = 1$. By the orthogonality of additive character and Lemma 2.3, we have

This completes the proof of the Theorem 1.7.

Example 4.1. We take $\mathbb{F}_{31}: = \{\overline{0}, \overline{1}, \cdots, \overline{30}\}$. Consider the cubic equations $x_1^3+2x_2^3+3x_3^3\equiv 0 (\bmod 31)$ and $x_1^3+2x_2^3\equiv3 (\bmod 31)$.

If the integers $c$ and $d$ satisfying that $4\cdot 31 = c^2+27d^2, c\equiv 1 (\bmod 3), 9d\equiv c(2\times6^{\frac{31-1}{3}}+1)(\bmod 31)$, then $c = 4, d = 2$. One can check that $2^{\frac{31-1}{3}}\equiv 1 (\bmod 31)$ and $6^{\frac{31-1}{3}}\equiv 25 (\bmod 31)$, so 6 is not a cubic element in $\mathbb{F}_{31}$ and 2 is a cubic element in $\mathbb{F}_{31}$. Thus $6\notin H$ and $1\equiv 2 (\bmod H)$.

It then follows from Theorems 1.4 and 1.5 that the numbers $M(1, 2, 3)$ and $N(1, 2, 3)$ of the cubic equations $x_1^3+2x_2^3+3x_3^3\equiv 0 (\bmod 31)$ and $x_1^3+2x_2^3\equiv3 (\bmod 31)$ are given by

and

We list the solutions of equation $x_1^3+2x_2^3\equiv3 (\bmod 31)$ as belove:

Acknowledgments

The authors are partially supported by the National Natural Science Foundation of China (Grant No. 11871193, 12071132) and the Natural Science Foundation of Henan Province (No. 222300420493, 202300410031).