scholarly journals Pythagorean triples and quadratic residues modulo an odd prime

2021 ◽  
Vol 7 (1) ◽  
pp. 957-966
Author(s):  
Jiayuan Hu ◽  
◽  
Yu Zhan ◽  
Keyword(s):  
Character Sums ◽  
Pythagorean Triples ◽  
Quadratic Residues ◽  
Elementary Methods ◽  
Modulo P

<abstract><p>In this article, we use the elementary methods and the estimate for character sums to study a problem related to quadratic residues and the Pythagorean triples, and prove the following result. Let $ p $ be an odd prime large enough. Then for any positive number $ 0 &lt; \epsilon &lt; 1 $, there must exist three quadratic residues $ x, \ y $ and $ z $ modulo $ p $ with $ 1\leq x, \ y, \ z\leq p^{1+\epsilon} $ such that the equation $ x^2+y^2 = z^2 $.</p></abstract>

scholarly journals The Legendre’s Symbol Modulo p and Its Some Elementary Results

2021 ◽  
Vol 2021 ◽  
pp. 1-6
Author(s):  
Jianghua Li ◽  
Yuan Zhang
Keyword(s):  
Character Sums ◽  
Asymptotic Formulas ◽  
Elementary Methods ◽  
Modulo P ◽  
Congruence Equation

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 p (an odd prime) and give some interesting identities and asymptotic formulas for it.

scholarly journals On Pythagorean Triples and the Primitive Roots Modulo a Prime

2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Jingzhe Wang
Keyword(s):  
Character Sums ◽  
Pythagorean Triples ◽  
Elementary Methods ◽  
Primitive Roots

In this paper, we use the elementary methods and the estimates for character sums to study a problem related to primitive roots and the Pythagorean triples and prove the following result: let p be an odd prime large enough. Then, there must exist three primitive roots x ,   y , and z modulo p such that x 2 + y 2 = z 2 .

scholarly journals The Mean Values of Character Sums and Their Applications

Mathematics ◽  
2021 ◽  
Vol 9 (4) ◽  
pp. 318
Author(s):  
Jiafan Zhang ◽  
Yuanyuan Meng
Keyword(s):  
Character Sums ◽  
Gauss Sums ◽  
Mean Values ◽  
Special Polynomials ◽  
Elementary Methods ◽  
The Mean

In this paper, we use the elementary methods and properties of classical Gauss sums to study the calculation problems of some mean values of character sums of special polynomials, and obtained several interesting calculation formulae for them. As an application, we give a criterion for determining that 2 is the cubic residue for any odd prime p.

scholarly journals The Quadratic Residues and Some of Their New Distribution Properties

Symmetry ◽  
2020 ◽  
Vol 12 (3) ◽  
pp. 421 ◽  
Author(s):  
Tingting Wang ◽  
Xingxing Lv
Keyword(s):  
Counting Function ◽  
Character Sums ◽  
Asymptotic Formulas ◽  
Open Problems ◽  
Quadratic Residues ◽  
New Distribution ◽  
Mod P

In this paper, we give some interesting identities and asymptotic formulas for one kind of counting function, by studying the computational problems involving the symmetry sums of one kind quadratic residues and quadratic non-residues mod p . The main methods we used are the properties of the Legendre’s symbol mod p , and the estimate for character sums. As application, we solve two open problems proposed by Zhiwei Sun.

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

2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Yuanyuan Meng
Keyword(s):  
Character Sums ◽  
Gauss Sums ◽  
Quadratic Character ◽  
Elementary Methods ◽  
Mod P

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.

scholarly journals The hybrid power mean of some special character sums of polynomials and two-term exponential sums modulo $ p $

2021 ◽  
Vol 6 (10) ◽  
pp. 10989-11004
Author(s):  
Wenpeng Zhang ◽  
◽  
Jiafan Zhang ◽  
Keyword(s):  
Exponential Sums ◽  
Character Sums ◽  
Gauss Sums ◽  
Asymptotic Formulas ◽  
Analytic Method ◽  
Power Mean ◽  
Hybrid Power ◽  
Special Character ◽  
Modulo P ◽  
Hybrid Power Mean

<abstract><p>We consider the calculation problem of one kind hybrid power mean involving the character sums of polynomials and two-term exponential sums modulo $ p $, an odd prime, and use the analytic method and the properties of classical Gauss sums to give some identities and asymptotic formulas for them.</p></abstract>

scholarly journals On the Quartic Residues and Their New Distribution Properties

Mathematics ◽  
2020 ◽  
Vol 8 (8) ◽  
pp. 1337
Author(s):  
Juanli Su ◽  
Jiafan Zhang
Keyword(s):  
Asymptotic Formula ◽  
Fourth Order ◽  
Character Sums ◽  
Exact Calculation ◽  
Calculation Formula ◽  
Analytic Methods ◽  
Counting Functions ◽  
Modulo P ◽  
New Distribution

In this paper, we use the analytic methods, the properties of the fourth-order characters, and the estimate for character sums to study the computational problems of one kind of special quartic residues modulo p, and give an exact calculation formula and asymptotic formula for their counting functions.

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