Erdős, Paul; Sárközy, A. On divisibility properties of integers of the form \(a+a'\). (English) Zbl 0625.10038 Acta Math. Hung. 50, 117-122 (1987). 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. Reviewer: K.Ramachandra Cited in 2 ReviewsCited in 5 Documents MSC: 11N35 Sieves 11B83 Special sequences and polynomials Keywords:arithmetic properties; sums of sequences of integers; square-free integers; \(k\)-free integers; large sieve PDFBibTeX XMLCite \textit{P. Erdős} and \textit{A. Sárközy}, Acta Math. Hung. 50, 117--122 (1987; Zbl 0625.10038) Full Text: DOI References: [1] A. Balog and A. Sárközy, On sums of sequences of integers.II,Acta Math. Hung.,44 (1984), 339–349. [2] P. Erdös and A. Sárközy, On differences and sums of integers,I,J. Number Theory,10 (1978), 430–450. · Zbl 0404.10029 [3] H. L. Montgomery,Topics in Multiplicative Number Theory, Springer Verlag (1971). · Zbl 0216.03501 [4] A. Sárközy and C. L. Stewart, On divisors of sums of integers,II,J. Reine Angew. Math., to appear. · Zbl 0649.10042 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.