On a Certain Quadratic Character Sums of Ternary Symmetry Polynomials modp

Yuanyuan Meng School of Mathematics, Northwest University, Xi’an, Shaanxi, China Correspondence should be addressed to Yuanyuan Meng; yymeng@stumail.nwu.edu.cn Received 1 March 2021; Accepted 31 March 2021; Published 17 April 2021 Academic Editor: Tingting Wang Copyright © 2021 Yuanyuan Meng. &is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 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 p and obtain some interesting identities for them.


Introduction
Let p be an odd prime, ( * /p) denotes the Legendre symbol mod p, i.e., for any integer n, one has n p � 1, if n is a quadratic residue mod p, −1, if n is a quadratic nonresidue mod p, Some of the most commonly used properties of the Legendre symbol are as follows (see [1,2]): where p and q are two different odd primes. e 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 p is a prime with p ≡ 1mod4, then for any integers r and s with (rs/p) � −1, one has the identity (see eorems 4-11 in [2]) p � α 2 (p) + β 2 (p) � (p− 1)/2 a�1 a + ra p From (3), we naturally wonder, for other forms of primes p, can they also be expressed in terms of Legendre's symbol?
In particular, if p is an odd prime with p ≡ 1mod3, then there are two integers d and b such that the identity (see [3]) where d is uniquely determined by d ≡ 1mod3. In addition, if p is an odd with p ≡ 3mod4, then there are two integers x and y such that Although we have not found the representations of d and b or x and y in terms of the Legendre symbol modulo p, we found that a certain quadratic character sum of the ternary symmetry polynomials are closely related to the numbers d and b.
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 p and obtain several interesting identities for them.
at is, we shall prove the following results.
Theorem 2. Let p be an odd prime with p ≡ 1mod6. If 2 is a cubic residue modulo p, then we have the identity where d is the same as defined in (4).
From these theorems, we may immediately deduce the following three corollaries.
Notes: it is clear that our method is applicable to multivariate symmetry polynomials f(x 1 , x 2 , . . . , x k ). But, when k is larger, the calculation is more complicated, so we do not give it. eorem 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.

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. ey 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][5][6][7][8][9][10][11]; here, we do not need to list.

Lemma 1.
Let p be a prime with p ≡ 1mod3. en, for any third-order character λmodp, one has the identity where d is the same as defined in (4).
Proof. See the work of Zhang and Hu [12]. □ Lemma 2. Let p be a prime with p ≡ 1mod6. en, for any sixth-order character ψmodp, one has the identity where i 2 � −1 and d is the same as defined in (4).
Proof. is result is Lemma 3 in the work of Chen [4], so we omit the proof process. □ Lemma 3. Let p be a prime with p ≡ 1mod6. en for any third-order character λmodp, we have the identity where ( * /p) � χ 2 denotes the Legendre symbol modulo p.
Proof. Note that χ 2 2 � χ 0 and τ 2 (χ 2 ) � χ 2 (−1) · p; from the properties of Gauss sums and the Legendre symbol modulo p, we have It is clear that λ(−1) � 1 and and from the properties of the Gauss sums, we can get On the other hand, note that λ 2 � λ; we also have

Journal of Mathematics
Taking the conjugate given above, we can deduce the other identity. is proves Lemma 5. □ Lemma 6. Let p be a prime with p ≡ 1 mod 6. en, for any third-order character λ mod p, we have the identity Proof. From (18) Combining (35)-(37), we have is proves Lemma 6.

Proofs of the Theorems
Now, we prove our theorems. From the properties of the reduced residue system modulo p, we have If (3, p − 1) � 1, when a passes through a reduced residue system modulo p, then a 3 also passes through a reduced residue system modulo p. So, from (39) and (15) (40) is proves eorem 1. If p ≡ 1 mod 6, let λ be a third-order character modulo p; then, for any integer a with (a, p) � 1, we have the identity