Fundamental solutions to Pell equation with prescribed size
type: Item
instance of: scholarly article
title: Fundamental solutions to Pell equation with prescribed size
zbMATH Open document ID: 1309.11023
DOI: 10.1134/S0081543812010051
author: Étienne Fouvry
author: Florent Jouve
published in: Proceedings of the Steklov Institute of Mathematics
publication date: 2014-09-17T00:00:00Z
review text: Let \(\varepsilon_D=T+U\sqrt D>1\) be the fundamental solution to the Pell equation \(T^2-DU^2=1\) and let \[ S^f(x;\alpha_1,\alpha_2):=\sharp \{D; \; D\leq x, D\neq \square, D^{1/2+\alpha_1}\leq \varepsilon_D \leq D^{1/2+\alpha_2} \}. \] \textit{C. Hooley} [J. Reine Angew. Math. 353, 98--131 (1984; Zbl 0539.10019)] conjectured that for every pair of real numbers \((\alpha_1,\alpha_2)\) such that \(0\leq \alpha_1<\alpha_2\) there exists \(c(\alpha_1,\alpha_2)>0\) and \(X(\alpha_1,\alpha_2)\) such that, \[ S^f(x;\alpha_1,\alpha_2)\geq c(\alpha_1,\alpha_2)x^{1/2} (\log x)^2 \] provided \(x\geq X(\alpha_1,\alpha_2)\). The main result of the paper under review is that the conjecture is true if \(\alpha_1<3/2\). The main ingredient in the proof is to transform the problem into a suitable character sum and use the famous Polya-Vinogradov inequality to estimate this sum.
reviewed by: Volker Ziegler
Mathematics Subject Classification ID: 11D09
zbMATH DE Number: 6345303
zbMATH Keywords: Pell equation
zbMATH Keywords: fundamental solution
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 6345303