Published online by Cambridge University Press: 11 October 2018
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
$D^{1/2+\unicode[STIX]{x1D6FC}}$ with
$\unicode[STIX]{x1D6FC}\in [\frac{1}{2},1]$ 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.