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Abstract

Let $d\geq 2$ be a squarefree integer, let $\omega \in \big \{\sqrt {d},\frac {1+\sqrt {d}}{2}\big \}$ be such that $\mathbb {Z}[\omega ]$ is the ring of algebraic integers of the real quadratic number field $\mathbb {Q}(\sqrt {d})$, let $\varepsilon \gt 1$ 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.