An elementary remark on the distribution of integers representable by a positive-definite integral binary quadratic form
type: Item
instance of: scholarly article
title: An elementary remark on the distribution of integers representable by a positive-definite integral binary quadratic form
zbMATH Open document ID: 0819.11014
DOI: 10.1007/BF01200436
author: Pierre Kaplan
author: Kenneth S. Williams
published in: Archiv der Mathematik
publication date: 1995-04-18T00:00:00Z
review text: Let \(f(X,Y)= aX^ 2+ bXY+ cY^ 2\) be a positive-definite integral binary quadratic form of discriminant \(-\Delta= b^ 2- 4ac<0\). Let \(m_ 1\) be the least positive integer represented by \(f\). Then, for every integer \(n\geq m_ 1\), there exist integers \(x\) and \(y\) such that \[ n< ax^ 2+ bxy+ cy^ 2< n+ 2m_ 1^{1/4} \Delta^{1/4} n^{1/4}+ m_ 1. \] This is proved in an elementary way. For \(f(X, Y)= aX^ 2+ cY^ 2\) and \(n\geq {{c^ 3}\over {a^ 2}}\) this is further improved, generalizing a theorem of \textit{S. Uchiyama} [J. Fac. Sci., Hokkaido Univ., I. Ser. 18, 124-127 (1964; Zbl 0199.087)].
Mathematics Subject Classification ID: 11E16
Mathematics Subject Classification ID: 11E12
zbMATH DE Number: 743528
zbMATH Keywords: distribution of integers represented by quadratic forms
zbMATH Keywords: positive- definite integral binary quadratic form
reviewed by: Meinhard Peters
MaRDI profile type: Publication
cites work: Q5611106
cites work: Q5596959
cites work: A Note on the Quadratic Form ax2 + 2bxy + cy2
full work available at URL: https://doi.org/10.1007/bf01200436
OpenAlex ID: W2121208443
description: scientific article; zbMATH DE number 743528