  Let $p$ be an odd prime, and $m,r \in \mathbb{Z}^+$ with $m$ coprime to $p$.
In this paper we investigate the real quadratic fields $K =
\mathbb{Q}(\sqrt{m^2p^{2r} + 1})$. We first show that for $m < C$, where
constant $C$ depends on $p$, the fundamental unit $\varepsilon$ of $K$
satisfies the congruence $\varepsilon^{p-1} \equiv 1 \mod{p^2}$, which implies
that $K$ is a non $p$-rational field. Varying $r$ then gives an infinite family
of non $p$-rational fields. When $m = 1$ and $p$ is a non-Wieferich prime, we
use a criterion of Fukuda and Komatsu to show that if $p$ does not divide the
class number of $K$, then the Iwasawa invariants for cyclotomic
$\mathbb{Z}_p$-extension of $K$ vanish. We conjecture that there are infinitely
many $r$ such that $p$ does not divide the class number of $K$.
