On divisibility properties of integers of the form \(a+a'\)
type: Item
instance of: scholarly article
title: On divisibility properties of integers of the form \(a+a'\)
zbMATH Open document ID: 0625.10038
DOI: 10.1007/BF01903370
author: András Sárközy
author: P. Erdős
published in: Acta Mathematica Hungarica
publication date: 1987-01-01T00:00:00Z
review text: Let \(A\subseteq \{1,2,...,N\}\) be such that \(a+a'\) for \(a\in A\), \(a'\in A\) be all square-free. Then the authors prove the following interesting results. Theorem 1. For \(N>N_0\) there exists an \(A\) such that \(|A| > (1/248)\log N.\) Theorem 2. For \(N>N_1\) every \(A\) satisfies \(|A| < 3N^{3/4} \log N.\) The proofs depend upon complicated applications of the large sieve and nice ingenuity which is characteristic of the authors. They remark that by slightly more complicated methods, they can get analogous results for \(k\)-free numbers. They remark that they can also consider the following problem. Let \(A\subseteq \{1,2,...,N\}\) and \(B\subseteq \{1,2,...,N\}\) such that \(a+b\) is square-free for all \(a\in A\) and \(b\in B\). Then for \(N\geq N_2\), \(|A| |B| < N^{3/2+\epsilon}.\) They can show that \(|A| |B| /N\to \infty\) is possible. They also remark that there is an absolute positive constant c such that \(|A| > cN\), \(|B|\to\infty\) is possible.
Mathematics Subject Classification ID: 11N35
Mathematics Subject Classification ID: 11B83
zbMATH DE Number: 4014831
zbMATH Keywords: arithmetic properties
zbMATH Keywords: sums of sequences of integers
zbMATH Keywords: square-free integers
zbMATH Keywords: \(k\)-free integers
zbMATH Keywords: large sieve
MaRDI profile type: Publication
cites work: On differences and sums of integers. I
cites work: Topics in multiplicative number theory
cites work: On Divisors of Sums of Integers IV
full work available at URL: https://doi.org/10.1007/bf01903370
OpenAlex ID: W2041253229
description: scientific article; zbMATH DE number 4014831