A short note on the divisibility of class numbers of real quadratic fields

TL;DR

This paper establishes a lower bound on the count of real quadratic fields with discriminants divisible by certain primes and class numbers divisible by specific powers of 2 and 3, extending understanding of class number divisibility properties.

Contribution

It proves a new lower bound for the number of real quadratic fields with prescribed prime divisibility in discriminants and class numbers, using a method adapted from prior work.

Findings

Lower bound of $N_{3,l}(X) o ext{grows at least as } X^{7/8}$ for large X.

Demonstrates divisibility properties of class numbers in real quadratic fields.

Extends previous results on class number divisibility by primes.

Abstract

For any integer $l\geq 1$, let $p_1, p_2, \ldots, p_{l+2}$ be distinct prime numbers $\geq 5.$ For all real numbers $X>1,$ we let $N_{3,l}(X)$ denote the number of real quadratic fields $K$ whose absolute discriminant $d_K\leq X$ and $d_K$ is divisible by $(p_1\ldots p_{l+2})$ together with the class number $h_K$ of $K$ divisible by $2^{l}\cdot 3.$ Then, in this short note, by following the method in \cite{Byeonkoh}, we prove that $N_{3,l}(X) \gg X^\frac{7}{8}$ for all large enough $X$'s.