A Conjecture Connected with Units of Quadratic Fields

TL;DR

This paper investigates properties of units in quadratic fields, providing numerical data on the minimal powers of fundamental units contained in specific orders, and suggests asymptotic behavior of related ratios as parameters grow.

Contribution

It offers new numerical insights into the behavior of fundamental units in quadratic fields and conjectures about the limiting distribution of certain ratios involving primes and units.

Findings

Numerical data on minimal powers of fundamental units in quadratic orders.

Evidence suggesting the ratios rac{p\u00b11}{2n(p)} and rac{p\u00b11}{n(p)} approach limits.

Conjecture on the asymptotic distribution of these ratios as parameters tend to infinity.

Abstract

In this article, we consider the order $\mathcal{O}_{f}={x+yf\sqrt{d}:x,\ y \in \Z}$ with conductor $f\in\N$ in a real quadratic field $K=\mathbb{Q}(\sqrt{d})$ where $d>0$ is square-free and $d\equiv2,3\pmod 4$. We obtain numerical information about $ n(f)=n(p)=min{\nu\in\N : \varepsilon^{\nu}\in \mathcal{O}_{p}}$ where $\varepsilon>1$ is the fundamental unit of $K$ and $p$ is an odd prime. Our numerical results suggest that the frequencies of $\frac{p\pm1}{2n(p)}$ or $\frac{p\pm1}{n(p)}$ should have a limit as the ranges of $d$ and $p$ go to infinity.