On product representation of powers. I
type: Item
instance of: scholarly article
title: On product representation of powers. I
zbMATH Open document ID: 0840.11010
DOI: 10.1016/0195-6698(95)90039-X
author: Vera T. Sós
author: András Sárközy
author: P. Erdős
published in: European Journal of Combinatorics
publication date: 1996-03-27T00:00:00Z
review text: The authors study the solvability of the equation (1) \(a_1 a_2\cdots a_k= z^2\), \(a_1, a_2,\dots, a_k\in A\), \(a_1< a_2<\cdots< a_k\), \(x\in \mathbb{N}\) for fixed \(k\) and dense sets \(A\) of natural numbers. It is shown that if \(k\) is even, \(k\geq 4\), and if \(A\) is of positive density then the above equation is solvable. In particular, it is proved that if \[ \begin{aligned} F_k(n) & := \max_{A\subseteq [1,n]; (1)\text{ is not solvable for } A}| A|, \text{ and }\\ L_k(n) & := \max_{A\subseteq [1,n]; (1)\text{ is not solvable for }A} \sum_{a\in A} {1\over a}\end{aligned} \] then for all \(n\in \mathbb{N}\), \(F_2(n)\) is equal to the number of square-free integers not exceeding \(n\), that is \(F_2(n)\sim 6/\pi^2\cdot n\). Non-trivial upper and lower bounds for \(F_k(n)\) for \(k= 3,4,4j, 4j+ 2\) are established and it is shown that \((\log 2-\varepsilon)n < F_{2k+ 1} < n-(1- \varepsilon) n(\log n)^{-2}\) (the authors remark that the above lower bound could be improved slightly by a lengthy computation). Moreover, the following results on \(L_k\) are established for \(n\to \infty\): \(L_{4j} = (1+ o(1))\log\log n\), \(L_{4j+ 2} = (3/2+ o(1))\log\log n\), \(L_{2j+ 1} = 1+(1/2+ o(1))\log n\).
reviewed by: Martin Helm
Mathematics Subject Classification ID: 11B83
zbMATH DE Number: 823417
zbMATH Keywords: hybrid problems
zbMATH Keywords: product representation of squares
Wikidata QID: Q105956190
MaRDI profile type: Publication
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full work available at URL: 0195-6698(95)90039-x
OpenAlex ID: W2017786676
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description: scientific article; zbMATH DE number 823417