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

TL;DR

This paper explores conjectures about how class numbers are distributed among real quadratic fields generated by prime radicands, proposing probabilistic models and revealing potential hierarchical structures within these fields.

Contribution

It introduces conjectural probability formulas for class numbers in real quadratic fields and connects these to Cohen-Lenstra heuristics, suggesting intrinsic hierarchical organization.

Findings

Proposes probability formulas for class number distributions.

Links distribution patterns to Cohen-Lenstra heuristics.

Suggests natural hierarchical structures in real quadratic 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.