A counterexample to the Pellian equation conjecture of Mordell

Abstract

Let $d\geq 2$ be a squarefree integer, let $ω\in\{\sqrt{d},\frac{1+\sqrt{d}}{2}\}$ be such that $\mathbb{Z}[ω]$ is the ring of algebraic integers of the real quadratic number field $\mathbb{Q}(\sqrt{d})$, let $\varepsilon>1$ be the fundamental unit of $\mathbb{Z}[ω]$ and let $x$ and $y$ be the unique nonnegative integers with $\varepsilon=x+yω$. 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.