Counting fundamental solutions to the Pell equation with prescribed size. (11th November 2018)

• Record Type: Journal Article
• Authors: Xi, Ping
• Is Part Of: Compositio mathematica. Volume 154:Number 11(2018)
• Page Start: 2379
• Page End: 2402
• Publication Date: 2018-11-11
• Subjects: 11D09, -- 11N37, -- 11L07, -- 11T23 (primary); Pell equation, -- exponential sums, -- q-analogue of van der Corput method; Mathematics -- Periodicals; 510
• DOI: 10.1112/S0010437X18007480
• Languages: English
• ISSNs: 0010-437X

Abstract

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.