A simple proof of the infinitude of primes

IF 0.1 Q4 MATHEMATICS Elemente der Mathematik Pub Date : 2020-04-08 DOI:10.4171/em/407
F. Lemmermeyer
{"title":"A simple proof of the infinitude of primes","authors":"F. Lemmermeyer","doi":"10.4171/em/407","DOIUrl":null,"url":null,"abstract":"The following is a simplification of the proof of the infinitude of primes using continued fractions given by Barnes [1]. Assume that there are only finitely many prime numbers, namely 2, p1 = 3, . . . , pn. Let q = p1 · · · pn be the product of all odd primes; then q + 1 is not divisible by any odd prime, hence must be a power of 2. Since q +1 ≡ 2 mod 4, we must have q +1 = 2 and therefore q = 1: contradiction. Since no odd prime p ≡ 3 mod 4 can divide q + 1, the proof actually shows that there are infinitely many primes p ≡ 1 mod 4.","PeriodicalId":41994,"journal":{"name":"Elemente der Mathematik","volume":"75 1","pages":"80-80"},"PeriodicalIF":0.1000,"publicationDate":"2020-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/em/407","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Elemente der Mathematik","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.4171/em/407","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0

Abstract

The following is a simplification of the proof of the infinitude of primes using continued fractions given by Barnes [1]. Assume that there are only finitely many prime numbers, namely 2, p1 = 3, . . . , pn. Let q = p1 · · · pn be the product of all odd primes; then q + 1 is not divisible by any odd prime, hence must be a power of 2. Since q +1 ≡ 2 mod 4, we must have q +1 = 2 and therefore q = 1: contradiction. Since no odd prime p ≡ 3 mod 4 can divide q + 1, the proof actually shows that there are infinitely many primes p ≡ 1 mod 4.
查看原文
分享 分享
微信好友 朋友圈 QQ好友 复制链接
本刊更多论文
素数无穷大的简单证明
下面是对Barnes[1]给出的用连分式证明质数无穷的简化。假设只有有限个素数,即2,p1 = 3,…pn。设q = p1···pn为所有奇素数之积;那么q + 1不能被任何奇素数整除,因此它一定是2的幂。因为q +1≡2 mod 4,我们必须有q +1 = 2,因此q = 1:矛盾。因为没有奇数素数p≡3 mod 4能除q + 1,所以这个证明实际上表明有无穷多个素数p≡1 mod 4。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
求助全文
约1分钟内获得全文 去求助
来源期刊
自引率
0.00%
发文量
21
期刊最新文献
How many stones can we place in empty baskets? Aufgaben Relationship between the evolutionary development of a virus and logistic growth Integers expressible as sums of primes and composites A combinatorial approach for computing the determinants of the generalized Vandermonde matrices
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
现在去查看 取消
×
提示
确定
0
微信
客服QQ
Book学术公众号 扫码关注我们
反馈
×
意见反馈
请填写您的意见或建议
请填写您的手机或邮箱
已复制链接
已复制链接
快去分享给好友吧!
我知道了
×
扫码分享
扫码分享
Book学术官方微信
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术
文献互助 智能选刊 最新文献 互助须知 联系我们:info@booksci.cn
Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。
Copyright © 2023 Book学术 All rights reserved.
ghs 京公网安备 11010802042870号 京ICP备2023020795号-1