  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.
