{"title":"全等数的必要条件：pq 的素数 p≡1 (mod 8) 和 q≡3 (mod 8)","authors":"Shamik Das","doi":"10.1016/j.jnt.2024.04.011","DOIUrl":null,"url":null,"abstract":"<div><p>In this paper, we establish a crucial requirement for a number of the form <em>n</em>, having two prime factors <em>p</em> and <em>q</em> such that <span><math><mo>(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo>)</mo><mo>≡</mo><mo>(</mo><mn>1</mn><mo>,</mo><mn>3</mn><mo>)</mo><mspace></mspace><mo>(</mo><mrow><mi>mod</mi></mrow><mspace></mspace><mn>8</mn><mo>)</mo></math></span>, to qualify as a congruent number. Specifically, we present congruence relations modulo 16 for the 2-part of the class number of the imaginary quadratic field <span><math><mi>Q</mi><mo>(</mo><msqrt><mrow><mo>−</mo><mn>2</mn><mi>p</mi><mi>q</mi></mrow></msqrt><mo>)</mo></math></span> when <em>n</em> is congruent.</p></div>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-05-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Required condition for a congruent number: pq with primes p ≡ 1 (mod 8) and q ≡ 3 (mod 8)\",\"authors\":\"Shamik Das\",\"doi\":\"10.1016/j.jnt.2024.04.011\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>In this paper, we establish a crucial requirement for a number of the form <em>n</em>, having two prime factors <em>p</em> and <em>q</em> such that <span><math><mo>(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo>)</mo><mo>≡</mo><mo>(</mo><mn>1</mn><mo>,</mo><mn>3</mn><mo>)</mo><mspace></mspace><mo>(</mo><mrow><mi>mod</mi></mrow><mspace></mspace><mn>8</mn><mo>)</mo></math></span>, to qualify as a congruent number. Specifically, we present congruence relations modulo 16 for the 2-part of the class number of the imaginary quadratic field <span><math><mi>Q</mi><mo>(</mo><msqrt><mrow><mo>−</mo><mn>2</mn><mi>p</mi><mi>q</mi></mrow></msqrt><mo>)</mo></math></span> when <em>n</em> is congruent.</p></div>\",\"PeriodicalId\":0,\"journal\":{\"name\":\"\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0,\"publicationDate\":\"2024-05-21\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0022314X24001112\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0022314X24001112","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
摘要
在本文中，我们建立了一个关键的条件，即一个 n 形式的数，有两个质因数 p 和 q，且 (p,q)≡(1,3)(mod8), 才能被称为全等数。具体地说，我们提出了当 n 为全等数时，虚二次域 Q(-2pq) 的类数的 2 部分的 modulo 16 全等关系。
Required condition for a congruent number: pq with primes p ≡ 1 (mod 8) and q ≡ 3 (mod 8)
In this paper, we establish a crucial requirement for a number of the form n, having two prime factors p and q such that , to qualify as a congruent number. Specifically, we present congruence relations modulo 16 for the 2-part of the class number of the imaginary quadratic field when n is congruent.
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