形如8k+1、8k-1、8k+3和8k-3(k∈Z)的素数都有无穷多个  

Have infinitely many prime numbers in the forms of 8k+1,8k-1,8k+3 and 8k-3(k∈Z)

作　　者：陈川 宓玲 CHEN Chuan;MI Ling(Key Laboratory of Computing Power Network and Information Security,Ministry of Education,Shandong Computer Science Center(National Supercomputer Center in Jinan),Qilu University of Technology(Shandong Academy of Sciences),Jinan 250353,China;Shandong Provincial Key Laboratory of Industrial Network and Information System Security Shandong Fundamental Research Center for Computer Science,Jinan 250014,China;School of Mathematics and Statistics,Qilu University of Technology(Shandong Academy of Sciences),Jinan 250353,China)

机构地区：[1]齐鲁工业大学(山东省科学院)山东省计算中心(国家超级计算济南中心)算力互联网与信息安全教育部重点实验室,山东济南250353 [2]山东省工业网络和信息系统安全重点实验室山东省基础科学研究中心(计算机科学),山东济南250014 [3]齐鲁工业大学(山东省科学院)数学与统计学院,山东济南250353

出　　处：《齐鲁工业大学学报》2024年第5期76-80,共5页Journal of Qilu University of Technology

基　　金：山东省自然科学基金(ZR2021MF090);山东省科技型中小企业创新能力提升工程项目(2023TSGC0197);齐鲁工业大学(山东省科学院)人才科研项目(2023RCKY134)。

摘　　要：尝试利用反证法和分类讨论法等,分别给出了“形如8k+1(k∈Z)的素数有无穷多个”、“形如8k-1(k∈Z)的素数有无穷多个”、“形如8k+3(k∈Z)的素数有无穷多个”和“形如8k-3(k∈Z)的素数有无穷多个”的严格证明。所用知识都是初等数论中的基础知识,仅限于素数、整除、同余和Legendre符号的一些基本性质。为了证明主要结论,还首先推导出了两个很有用的引理。This article attempts to use the proof methods of reduction to absurdity and classification discussion to provide the strict proofs for“there are infinite primes in the form of 8k+1(k∈Z)”,“there are infinite primes in the form of 8k-1(k∈Z)”,“there are infinite primes in the form of 8k+3(k∈Z)”,and“there are infinite primes in the form of 8k-3(k∈Z)”.The used knowledge is the fundamental knowledge in elementary number theory,limited to some basic properties of prime number,integer division,congruence,and Legendre symbol.To prove the main conclusions,this article first derives two very useful lemmas.

关 键 词：反证法 素数 整除 同余 LEGENDRE符号 

分 类 号：O156.1[理学—数学]