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The cardinality of the set of $D\leqslant x$ for which the fundamentalsolution 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 certainlower bounds are obtained, improving previous results of Fouvry by introducingthe $q$-analogue of van der Corput method to algebraic exponential sums withsmooth moduli.

Counting fundamental solutions to the Pell equation with prescribed size