On the divisibility of the class numbers and discriminants of imaginary quadratic fields

Meng Fai Lim

Published 2016-01-20Version 1

Let $n$ be a squarefree positive odd integer. We will show that there exist infinitely many imaginary quadratic number fields with discriminant divisible by $n$ and-at the same time-having an element of order $n$ in the class group. We will also prove certain results on the divisibility of $r(N)$, where $r(N)$ is the representation numbers of $N$ as sums of three squares. Namely, we will show that for a given squarefree positive odd integer $n$ there exist infinitely many $N$ such that $n$ divides both $N$ and $r(N)$.