一个二元二次同余方程解的计数

doi:10.6040/j.issn.1671-9352.0.2018.407

《山东大学学报(理学版)》 ›› 2019, Vol. 54 ›› Issue (8): 108-120.doi: 10.6040/j.issn.1671-9352.0.2018.407

一个二元二次同余方程解的计数

段然

西北大学数学学院, 陕西西安 710127

出版日期:2019-08-20 发布日期:2019-07-03
作者简介:段然(1989— ),男,博士研究生,研究方向为解析数论指数和同余方程. E-mail:duan.ran@stumail.nwu.edu.cn

Counting solutions of a binary quadratic congruence equation

DUAN Ran

School of Mathematics, Northwest University, Xian 710127, Shaanxi, China

Online:2019-08-20 Published:2019-07-03

摘要/Abstract

摘要： 设n是任意正整数,令Z_n是模n的剩余类环,并且Z^*_n是模n的即约剩余类环,即Z^*_n={s:1≤s≤n, gcd(s,n)=1}。通过利用同余理论与指数和的相关结果来研究集合T(a,b,c,n)={(x,y)∈(Z^*_n)²:ax²+by²+c≡0 mod n}的元素个数并给出集合T(a,b,c,n)元素个数的确切计算公式。

关键词: 同余方程, 剩余类环, 指数和, 集合划分

Abstract: Let n be a positive integer. Denote by Z_n the ring of residue classes mod n, and by Z^*_n the group of units in Z_n, i.e. Z^*_n={s:1≤s≤n and gcd(s,n)=1}. The main purpose of this paper is using congruence conclusion and some results of exponential sums to study the number of elements of the set T(a,b,c,n)={(x,y)∈(Z^*_n)2:ax2+by2+c≡0 mod n} and give an exact computational formula for the number of elements of T(a,b,c,n).

Key words: congruence equation, ring of residue class, exponential sum, set partition

中图分类号:

O156.4

段然. 一个二元二次同余方程解的计数[J]. 《山东大学学报(理学版)》, 2019, 54(8): 108-120.

DUAN Ran. Counting solutions of a binary quadratic congruence equation[J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2019, 54(8): 108-120.

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一个二元二次同余方程解的计数

Counting solutions of a binary quadratic congruence equation

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