Integers without large prime factors
type: Item
instance of: scholarly article
title: Integers without large prime factors
zbMATH Open document ID: 0797.11070
published in: Journal de Théorie des Nombres de Bordeaux
publication date: 1994-04-24T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/93590
review text: Since \textit{K. K. Norton}'s survey [Mem. Am. Math. Soc. 106, 106 p. (1971; Zbl 0211.37801)], there has been an explosion of interest in the distribution of integers without large prime factors, the so-called ``smooth numbers'', in large part because of the ground-breaking contributions of the present two authors, Alladi, Balog, Erdős, Friedlander, Hensley, Ivić, Pomerance and many others. Asymptotic estimates have now been proved, in very wide ranges, for the number of smooth numbers up to \(x\), in arithmetic progressions, in short intervals, as one less than a prime, \(\dots\). Moreover they have been applied to help our understanding of a wide diversity of number theory questions, such as Waring's problem, the Fermat and \(abc\) conjectures, the analysis of algorithms, particularly for factoring and primality testing, Carmichael numbers, gaps between primes, and a host of other problems besides. The authors here survey these and other developments in the subject. The survey is excellently organized, well-written, thorough, and goes into considerable depth for those questions which deserve that kind of attention. This will now complement Norton's survey as the primary references in the area; and it also will serve as an excellent guide for those who wish to learn about the subject.
reviewed by: Andrew Granville
Mathematics Subject Classification ID: 11N25
Mathematics Subject Classification ID: 11-02
zbMATH DE Number: 556386
zbMATH Keywords: smooth numbers
zbMATH Keywords: asymptotic estimates
zbMATH Keywords: distribution of integers without large prime factors
zbMATH Keywords: arithmetic progressions
zbMATH Keywords: short intervals
zbMATH Keywords: survey
author: Adolf Hildebrand
author: Gérald Tenenbaum
MaRDI profile type: Publication
OpenAlex ID: W2088772065
Wikidata QID: Q29037383
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Integers without large prime factors (DOI)
DOI: 10.5802/JTNB.101

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