On a Certain Quadratic Character Sums of Ternary Symmetry Polynomials  mod p

Yuanyuan Meng,Tingting Wang

doi:10.1155/2021/5572835

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 p and obtain some interesting identities for them.

Highlights

Let p be an odd prime, (∗/p) denotes the Legendre symbol mod p, i.e., for any integer n, one has n ⎧⎪⎪⎪⎪⎨ 1, if n is a quadratic residue mod p, 􏼠􏼡 p ⎪⎪⎪⎪⎩−1, if n is a quadratic nonresidue mod p, (1) 0, if p | n.Some of the most commonly used properties of the Legendre symbol are as follows: −1 􏼠􏼡 p (−1)(p− 1)/2,􏼠2􏼡 (−1)(p2− 1)/8
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
If p is an odd prime with p ≡ 1mod6, if 2 is not a cubic res􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌ipad􏽘 −u11epb􏽘 −m11 opc􏽘 d−11u􏼠loa4pb,c we have + b4ac + c4ab p

Summary

Introduction

Let p be an odd prime, (∗/p) denotes the Legendre symbol mod p, i.e., for any integer n, one has n ⎧⎪⎪⎪⎪⎨ 1, if n is a quadratic residue mod p,. 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. If 2 is not a cubic residue modulo p, we have the identity p−1 p−1 p−1 a4bc + b4ac + c4ab + abc. From these theorems, we may immediately deduce the following three corollaries. If p is an odd prime with p ≡ 1mod, if 2 is not a cubic res􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌ipad􏽘 −u11epb􏽘 −m11 opc􏽘 d−11u􏼠loa4pb,c we have + b4ac + c4ab p. Interested readers are encouraged to join us in the research

Several Lemmas

Findings

Proofs of the Theorems