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arXiv · 2508.08447

Locally Associated Orders in Real Quadratic Number Fields

Grant Moles·Talha Khan
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Abstract

In 2025, the concept of an order in a number field being associated, ideal-preserving, or locally associated was introduced in order to tackle problems in factorization. In this paper, we explore locally associated orders in real quadratic number fields of the form $\mathbb{Q}[\sqrt{p}]$, with $p\in\mathbb{N}$ prime. In particular, we develop strategies and produce results which make determining when a given order in such a number field is (or is not) locally associated much easier. We also highlight the relatively few cases which defy simple characterization, leading to a conjecture on the solutions to Pell's equations of the form $x^2-y^2p=1$.

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Grant Moles, Talha Khan. 2025-12-02. Locally Associated Orders in Real Quadratic Number Fields. https://arxiv.org/abs/2508.08447

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