A Conjecture Connected with Units of Quadratic Fields

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.