  The cardinality of the set of $D\leqslant x$ for which the fundamental
solution of the Pell equation $t^2-Du^2=1$ is less than
$D^{\frac{1}{2}+\alpha}$ with $\alpha\in[\frac{1}{2},1]$ is studied and certain
lower bounds are obtained, improving previous results of Fouvry by introducing
the $q$-analogue of van der Corput method to algebraic exponential sums with
smooth moduli.
