NOTE ON THE CLASS NUMBER OF IMAGINARY QUADRATIC NUMBER FIELDS AND SOPHIE GERMAIN PRIMES

IF 0.2 Q4 MATHEMATICS JP Journal of Algebra Number Theory and Applications Pub Date : 2023-11-06 DOI:10.17654/0972555523027

Anly Li

引用次数: 0

Abstract

Let $p>2$ be a Sophie Germain prime which means that $q=2 p+1$ is also a prime. Let $K=\mathbb{Q}(\sqrt{-q})$ and denote its ideal class number by $h_K$. We use Kummer congruences and Bernoulli numbers to study the relation between $p$ and $h_K$. We prove that $-h_K \equiv k \cdot p(\bmod q)$ for some positive integer $k$. Received: August 29, 2023Accepted: October 7, 2023

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关于虚二次数域的类数和苏菲日耳曼素数的注意事项

假设$p>2$是Sophie Germain质数，也就是说$q=2 p+1$也是质数。设$K=\mathbb{Q}(\sqrt{-q})$，并用$h_K$表示它的理想类数。我们利用Kummer同余和Bernoulli数研究了$p$和$h_K$之间的关系。我们证明了$-h_K \equiv k \cdot p(\bmod q)$对于某个正整数$k$。收稿日期:2023年8月29日。收稿日期:2023年10月7日

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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JP Journal of Algebra Number Theory and Applications MATHEMATICS-

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期刊介绍： The JP Journal of Algebra, Number Theory and Applications is a peer-reviewed international journal. Original research papers theoretical, computational or applied, in nature, in any branch of Algebra and Number Theory are considered by the JPANTA. Together with the core topics in these fields along with their interplay, the journal promotes contributions in Diophantine equations, Representation theory, and Cryptography. Realising the need of wide range of information for any emerging area of potential research, the journal encourages the submission of related survey articles as well.