## The infinitude of $\mathbb {Q}(\sqrt {-p})$ with class number divisible by 16

### Volume 178 / 2017

#### Abstract

The density of primes $p$ such that the class number $h$ of $\mathbb{Q}(\sqrt{-p})$ is divisible by $2^k$ is conjectured to be $2^{-k}$ for all positive integers $k$. The conjecture has been proved for $1\leq k\leq 3$. For $k\geq 4$, however, it is still open and a similar approach via Chebotarev’s density theorem does not appear to be possible. For primes $p$ of the form $p = a^2 + c^4$ with $c$ even, we describe the 8-Hilbert class field of $\mathbb{Q}(\sqrt{-p})$ in terms of $a$ and $c$. We then adapt a theorem of Friedlander and Iwaniec to show that there are infinitely many primes $p$ for which $h$ is divisible by $16$, and also infinitely many primes $p$ for which $h$ is divisible by $8$ but not by $16$.