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Counting fundamental solutions to the Pell equation with prescribed size

Part of: Diophantine equations Exponential sums and character sums Finite fields and commutative rings (number-theoretic aspects) Multiplicative number theory

Published online by Cambridge University Press:  11 October 2018

Ping Xi
Ping Xi*
Affiliation:
School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an 710049, PR China email [email protected]
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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.

Keywords

Pell equationexponential sumsq-analogue of van der Corput method

MSC classification

Primary: 11D09: Quadratic and bilinear equations 11N37: Asymptotic results on arithmetic functions 11L07: Estimates on exponential sums 11T23: Exponential sums
Type
Research Article
Information
Compositio Mathematica , Volume 154 , Issue 11 , November 2018 , pp. 2379 - 2402
Copyright
© The Author 2018 

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