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

Abstract

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 then apply our result to prove that for a given squarefree positive odd integer $n$ there exist infinitely many $N$ such that $n$ divides both $N$ and $r(N)$, where $r(N)$ is the representation number of $N$ as sums of three squares.

arXiv commentProp 5.1 of the paper is incorrect. Furthermore, it has been pointed out to the author that the main results of the paper (Theorem 1.1) follow from classical results in literature