.svg){kind=logo}
NOTE ON THE CLASS NUMBER OF IMAGINARY QUADRATIC NUMBER FIELDS AND SOPHIE GERMAIN PRIMES
Keywords:
class number, Sophie Germain prime.DOI:
https://doi.org/10.17654/0972555523027Abstract
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, 2023
Accepted: October 7, 2023
References
Kenneth F. Ireland and Michael I. Rosen, A Classical Introduction to Modern Number Theory, Springer, 1990.
Paulo Ribenboim, 13 Lectures on Fermat’s Last Theorem, Springer-Verlag, New York, 1979.
Tom Lovering, Cyclotomic fields and Fermat’s last theorem.
https://tlovering.files.wordpress.com/2015/02/cyclotomic-fields-and-flt9.pdf.
Downloads
Published
Issue
Section
License
Copyright (c) 2023 PUSHPA PUBLISHING HOUSE, PRAYAGRAJ, INDIA
This work is licensed under a Creative Commons Attribution 4.0 International License.
_________________________________
Attribution: Credit Pusha Publishing House as the original publisher, including title and author(s) if applicable.
Non-Commercial Use: For non-commercial purposes only. No commercial activities without explicit permission.
No Derivatives: Modifying or creating derivative works not allowed without written permission.
Contact Pusha Publishing House for more info or permissions.
