arXiv subjects

Nihal Bircan

Publications and source records attributed to Nihal Bircan.

2 recordsLinked to original sources

arXiv2012

On the Power of Integers and Conductors of Quadratic Fields

We consider the integers $α$ of the quadratic field $ \mathbb{Q} (\sqrt{d}$ $)$ where $d\in \Z$ is square-free and $d\equiv 1,2,3 \pmod 4$. Let $p$ be an odd prime. Using the embedding into $ \text{GL}(2,\mathbb{Z})$ we obtain bounds for the first $ν\in\N$ such that $α^ν\equiv 1\pmod p.$ For the conductor $f$, we then study the first integer $n=n(f)$ such that $α^n\in\mathcal{O}_{f}$. We obtain bounds for $n(f)$ and for $n(fp^{k})$. The most interesting case is that $α$ is the fundamental unit of $ \mathbb{Q} (\sqrt{d}$ $)$.

math.NT↗

arXiv2012

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{ν\in\N : \varepsilon^ν\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.

math.NT↗