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

Abstract

For any integer $l\ge 1$, let $p_1,p_2,\dots ,p_{l+2}$ be distinct prime numbers $\ge 5.$ For all real numbers $X>1,$ we let $N_{3,l}(X)$ denote the number of real quadratic fields $K$ whose absolute discriminant $d_K\le X$ and $d_K$ is divisible by $(p_1\dots p_{l+2})$ 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.

arXiv commentTo appear in Journal of The Ramanujan Mathematical Society