{"title":"S.W. 格雷厄姆猜想的进展情况","authors":"S.V. Nagaraj","doi":"10.1016/j.jnt.2024.07.009","DOIUrl":null,"url":null,"abstract":"<div><p>This article describes progress towards a conjecture of S.W. Graham. He conjectured that the number <span><math><msub><mrow><mi>C</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo></math></span> of Carmichael numbers up to <em>X</em> with three prime factors is <span><math><mo>≤</mo><msqrt><mrow><mi>X</mi></mrow></msqrt></math></span> for all <span><math><mi>X</mi><mo>≥</mo><mn>1</mn></math></span>. He showed that his conjecture is true for <span><math><mi>X</mi><mo>≤</mo><msup><mrow><mn>10</mn></mrow><mrow><mn>16</mn></mrow></msup></math></span> and <span><math><mi>X</mi><mo>></mo><msup><mrow><mn>10</mn></mrow><mrow><mn>126</mn></mrow></msup></math></span>. In this article, it is shown that the conjecture is true for <span><math><mi>X</mi><mo>≤</mo><msup><mrow><mn>10</mn></mrow><mrow><mn>24</mn></mrow></msup></math></span> and <span><math><mi>X</mi><mo>></mo><mn>2</mn><mo>⁎</mo><msup><mrow><mn>10</mn></mrow><mrow><mn>40</mn></mrow></msup></math></span>. In both cases, analytical methods establish the conjecture for large <em>X</em> and tables of Carmichael numbers are used for small <em>X</em>.</p></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"266 ","pages":"Pages 281-294"},"PeriodicalIF":0.6000,"publicationDate":"2024-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0022314X24001756/pdfft?md5=a96d01f7aa1e622c98ed012747b85804&pid=1-s2.0-S0022314X24001756-main.pdf","citationCount":"0","resultStr":"{\"title\":\"Progress towards a conjecture of S.W. Graham\",\"authors\":\"S.V. Nagaraj\",\"doi\":\"10.1016/j.jnt.2024.07.009\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>This article describes progress towards a conjecture of S.W. Graham. He conjectured that the number <span><math><msub><mrow><mi>C</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo></math></span> of Carmichael numbers up to <em>X</em> with three prime factors is <span><math><mo>≤</mo><msqrt><mrow><mi>X</mi></mrow></msqrt></math></span> for all <span><math><mi>X</mi><mo>≥</mo><mn>1</mn></math></span>. He showed that his conjecture is true for <span><math><mi>X</mi><mo>≤</mo><msup><mrow><mn>10</mn></mrow><mrow><mn>16</mn></mrow></msup></math></span> and <span><math><mi>X</mi><mo>></mo><msup><mrow><mn>10</mn></mrow><mrow><mn>126</mn></mrow></msup></math></span>. In this article, it is shown that the conjecture is true for <span><math><mi>X</mi><mo>≤</mo><msup><mrow><mn>10</mn></mrow><mrow><mn>24</mn></mrow></msup></math></span> and <span><math><mi>X</mi><mo>></mo><mn>2</mn><mo>⁎</mo><msup><mrow><mn>10</mn></mrow><mrow><mn>40</mn></mrow></msup></math></span>. In both cases, analytical methods establish the conjecture for large <em>X</em> and tables of Carmichael numbers are used for small <em>X</em>.</p></div>\",\"PeriodicalId\":50110,\"journal\":{\"name\":\"Journal of Number Theory\",\"volume\":\"266 \",\"pages\":\"Pages 281-294\"},\"PeriodicalIF\":0.6000,\"publicationDate\":\"2024-08-20\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://www.sciencedirect.com/science/article/pii/S0022314X24001756/pdfft?md5=a96d01f7aa1e622c98ed012747b85804&pid=1-s2.0-S0022314X24001756-main.pdf\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Number Theory\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0022314X24001756\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Number Theory","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0022314X24001756","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
摘要
本文描述了在实现 S.W. Graham 猜想方面取得的进展。他猜想，对于所有 X≥1 的卡迈克尔数 C3(X)，直到 X 的三个质因数为 ≤X。他证明了他的猜想对于 X≤1016 和 X>10126 为真。本文则证明猜想在 X≤1024 和 X>2⁎1040 时为真。在这两种情况下，对于大的 X，都用分析方法建立了猜想，而对于小的 X，则使用了卡迈克尔数表。
This article describes progress towards a conjecture of S.W. Graham. He conjectured that the number of Carmichael numbers up to X with three prime factors is for all . He showed that his conjecture is true for and . In this article, it is shown that the conjecture is true for and . In both cases, analytical methods establish the conjecture for large X and tables of Carmichael numbers are used for small X.
期刊介绍：
The Journal of Number Theory (JNT) features selected research articles that represent the broad spectrum of interest in contemporary number theory and allied areas. A valuable resource for mathematicians, the journal provides an international forum for the publication of original research in this field.
The Journal of Number Theory is encouraging submissions of quality, long articles where most or all of the technical details are included. The journal now considers and welcomes also papers in Computational Number Theory.
Starting in May 2019, JNT will have a new format with 3 sections:
JNT Prime targets (possibly very long with complete proofs) high impact papers. Articles published in this section will be granted 1 year promotional open access.
JNT General Section is for shorter papers. We particularly encourage submission from junior researchers. Every attempt will be made to expedite the review process for such submissions.
Computational JNT . This section aims to provide a forum to disseminate contributions which make significant use of computer calculations to derive novel number theoretic results. There will be an online repository where supplementary codes and data can be stored.
京公网安备 11010802042870号
京ICP备2023020795号-1
分享
求助内容：
应助结果提醒方式：
扫码关注我们
