Conjectures on the distribution behavior of the class numbers of certain real quadratic number fields

Abstract

Given a random real quadratic field from $\{ \mathbb{Q}(\sqrt{p}\,) ~|~ p \text{ primes} \}$, the conjectural probability $\mathbb{P}(h=q)$ that it has class number $q$ is given for all positive odd integers $q$. Some related conjectures of the Cohen-Lenstra heuristic are given here as corollaries. These results suggest that the set of real quadratic number fields may have some natural hierarchical structures.