The largest prime factor of the integers in an interval
type: Item
instance of: scholarly article
title: The largest prime factor of the integers in an interval
zbMATH Open document ID: 0867.11064
author: D. R. Heath-Brown
published in: Science in China. Series A
publication date: 1997-08-10T00:00:00Z
review text: The main result in this paper is that the largest prime factor of integers in an interval of the form \([x,x+x^{1/2+\varepsilon}]\), with \(x\) sufficiently large, is at least \(x^{11/12-\varepsilon}\). The previously known best lower bound was \(x^{0.82}\) due to \textit{A. Balog}, \textit{G. Harman} and \textit{J. Pintz} [J. Lond. Math. Soc., II. Ser. 28, 218-226 (1983; Zbl 0514.10034)]. These authors used sieve methods, whereas the present method is more analytic, related to that of \textit{M. Jutila} [J. Indian Math. Soc., New Ser. 37 (1973), 43-53 (1974; Zbl 0317.10049)]. A further advance in this problem was made recently by \textit{G. Harman} [Sieve methods, exponential sums and their applications in number theory, Cambridge Univ. Press, 161-173 (1997)], who obtained the lower bound \(x^{19/20}\) by sieve methods.
Mathematics Subject Classification ID: 11N25
zbMATH DE Number: 978361
zbMATH Keywords: largest prime factor of integers in an interval
reviewed by: Matti Jutila
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 978361