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arXiv · 2402.09827

A counterexample to the Pellian equation conjecture of Mordell

Andreas Reinhart
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Abstract

Let $d\geq 2$ be a squarefree integer, let $\omega\in\{\sqrt{d},\frac{1+\sqrt{d}}{2}\}$ be such that $\mathbb{Z}[\omega]$ is the ring of algebraic integers of the real quadratic number field $\mathbb{Q}(\sqrt{d})$, let $\varepsilon>1$ be the fundamental unit of $\mathbb{Z}[\omega]$ and let $x$ and $y$ be the unique nonnegative integers with $\varepsilon=x+y\omega$. In this note, we extend and study the list of known squarefree integers $d\geq 2$, for which $y$ is divisible by $d$ (cf. OEIS A135735). As a byproduct, we present a counterexample to a conjecture of L. J. Mordell.

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Andreas Reinhart. 2024-02-15. A counterexample to the Pellian equation conjecture of Mordell. https://arxiv.org/abs/2402.09827

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