On sums with small prime factors
type: Item
instance of: scholarly article
title: On sums with small prime factors
zbMATH Open document ID: 0867.11069
DOI: 10.1007/BF01874496
author: Gábor N. Sárközy
published in: Acta Mathematica Hungarica
publication date: 1997-06-30T00:00:00Z
review text: It is proved that every sufficiently large integer \(N>N_0(\epsilon)\) (where \(\epsilon>0\) is fixed) can be written in the form \(N=n_1+n_2+n_3\) such that all prime factors of \(n_1n_2n_3\) are bounded by \[ \text{exp}\left(\big(\sqrt{3/2}+\epsilon\big)\big(\log N\log\log N\big)^{1/2}\right). \] This is an improvement on a result of \textit{A. Balog} and \textit{A. Sárközy} [Stud. Sci. Math. Hung. 19, 35-47 (1984; Zbl 0569.10025)] only in the constant in the exp function. However, it is also shown that the above result does not leave too much room for further improvements, at least not with the present known approach. The main innovation of the paper is the use of the large sieve rather than the Hardy--Littlewood method.
Mathematics Subject Classification ID: 11P32
Mathematics Subject Classification ID: 11N36
zbMATH DE Number: 868513
zbMATH Keywords: sums with small prime factors
zbMATH Keywords: representations of integers
zbMATH Keywords: large sieve
reviewed by: Antal Balog
MaRDI profile type: Publication
cites work: On additive representation of integers
cites work: On sums of sequences of integers, I
cites work: Q5803354
cites work: An additive problem in the theory of numbers
cites work: Q3737509
cites work: On the number of positive integers \(\leq x\) and free of prime factors \(>y\)
cites work: The analytic principle of the large sieve
cites work: Q4291376
cites work: On a problem of P. Erdős
cites work: Q3737510
description: scientific article; zbMATH DE number 868513