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2018•Compositio MathematicaOpen access

Counting fundamental solutions to the Pell equation with prescribed size

Ping Xi

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Abstract

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.

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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.

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Available abstract

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.

Key concepts: Mathematics, Cardinality (data modeling), Moduli, Algebraic number, Exponential function, Combinatorics, Moduli space, Discrete mathematics

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