关于非全等数作为非全等数的倍数

IF 0.8 2区数学 Q2 MATHEMATICS

Journal of Algebra Pub Date : 2025-09-17 DOI:10.1016/j.jalgebra.2025.09.006

Shenxing Zhang

{"title":"关于非全等数作为非全等数的倍数","authors":"Shenxing Zhang","doi":"10.1016/j.jalgebra.2025.09.006","DOIUrl":null,"url":null,"abstract":"<div><div>Let <span><math><mi>n</mi><mo>=</mo><mi>P</mi><mi>Q</mi></math></span> be a square-free positive integer, where <em>P</em> is a product of primes congruent to <span><math><mn>1</mn><mspace></mspace><mrow><mi>mod</mi></mrow><mspace></mspace><mn>8</mn></math></span>, and <em>Q</em> is a non-congruent number with a trivial 2-primary Shafarevich-Tate group. Under certain conditions on the Legendre symbols <span><math><mo>(</mo><mfrac><mrow><mi>q</mi></mrow><mrow><mi>p</mi></mrow></mfrac><mo>)</mo></math></span> for primes <span><math><mi>p</mi><mo>|</mo><mi>P</mi><mo>,</mo><mi>q</mi><mo>|</mo><mi>Q</mi></math></span>, we establish a criterion characterizing when <em>n</em> is non-congruent with a minimal or a second minimal 2-primary Shafarevich-Tate group. We also provide a sufficient condition for <em>n</em> to be non-congruent with a larger 2-primary Shafarevich-Tate group. These results involve the class groups and tame kernels of quadratic fields.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"687 ","pages":"Pages 394-418"},"PeriodicalIF":0.8000,"publicationDate":"2025-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On non-congruent numbers as multiples of non-congruent numbers\",\"authors\":\"Shenxing Zhang\",\"doi\":\"10.1016/j.jalgebra.2025.09.006\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><div>Let <span><math><mi>n</mi><mo>=</mo><mi>P</mi><mi>Q</mi></math></span> be a square-free positive integer, where <em>P</em> is a product of primes congruent to <span><math><mn>1</mn><mspace></mspace><mrow><mi>mod</mi></mrow><mspace></mspace><mn>8</mn></math></span>, and <em>Q</em> is a non-congruent number with a trivial 2-primary Shafarevich-Tate group. Under certain conditions on the Legendre symbols <span><math><mo>(</mo><mfrac><mrow><mi>q</mi></mrow><mrow><mi>p</mi></mrow></mfrac><mo>)</mo></math></span> for primes <span><math><mi>p</mi><mo>|</mo><mi>P</mi><mo>,</mo><mi>q</mi><mo>|</mo><mi>Q</mi></math></span>, we establish a criterion characterizing when <em>n</em> is non-congruent with a minimal or a second minimal 2-primary Shafarevich-Tate group. We also provide a sufficient condition for <em>n</em> to be non-congruent with a larger 2-primary Shafarevich-Tate group. These results involve the class groups and tame kernels of quadratic fields.</div></div>\",\"PeriodicalId\":14888,\"journal\":{\"name\":\"Journal of Algebra\",\"volume\":\"687 \",\"pages\":\"Pages 394-418\"},\"PeriodicalIF\":0.8000,\"publicationDate\":\"2025-09-17\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Algebra\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0021869325005319\",\"RegionNum\":2,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Algebra","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0021869325005319","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}

引用次数: 0

摘要

设n=PQ是一个无平方正整数，其中P是与1mod8相等的素数的乘积，并且Q是一个非同余数，具有平凡的2-初级shafarevic - tate群。在素数p| p，q| q的Legendre符号（qp）的一定条件下，我们建立了n与极小或次极小2-初等shafareich - tate群不同余的判据。我们还给出了n与一个更大的2-初等shafarevic - tate群不一致的充分条件。这些结果涉及到二次域的类群和驯服核。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

查看原文本刊更多论文

On non-congruent numbers as multiples of non-congruent numbers

Let

n = P Q

be a square-free positive integer, where P is a product of primes congruent to

1 \mod 8

, and Q is a non-congruent number with a trivial 2-primary Shafarevich-Tate group. Under certain conditions on the Legendre symbols

(\frac{q}{p})

for primes

p | P, q | Q

, we establish a criterion characterizing when n is non-congruent with a minimal or a second minimal 2-primary Shafarevich-Tate group. We also provide a sufficient condition for n to be non-congruent with a larger 2-primary Shafarevich-Tate group. These results involve the class groups and tame kernels of quadratic fields.

求助全文

通过发布文献求助，成功后即可免费获取论文全文。去求助

来源期刊

Journal of Algebra 数学-数学

CiteScore

1.50

自引率

22.20%

发文量

414

审稿时长

2-4 weeks

期刊介绍： The Journal of Algebra is a leading international journal and publishes papers that demonstrate high quality research results in algebra and related computational aspects. Only the very best and most interesting papers are to be considered for publication in the journal. With this in mind, it is important that the contribution offer a substantial result that will have a lasting effect upon the field. The journal also seeks work that presents innovative techniques that offer promising results for future research.