Remarks on a paper of J. Barát and P.P. Varjú

Bollobás, Béla

doi:10.5802/jtnb.1212

Remarks on a paper of J. Barát and P.P. Varjú

Bollobás, Béla ^1,²

¹ Department of Pure Mathematics and Mathematical Statistics, Wilberforce Road, Cambridge, CB3 0WA, UK
² Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152, USA

Journal de théorie des nombres de Bordeaux, Volume 34 (2022) no. 2, pp. 515-516.

Abstract
Résumé

Let ${d_{i} n + b_{i} : n \in ℤ}_{i \in I}$ be a family of disjoint arithmetic progressions covering the integers. Barát and Varjú [1] have proved that if $d_{i} = p_{1}^{α_{1}} p_{2}^{α_{2}}$ for two prime numbers $p_{1}$ , $p_{2}$ and integers des $α_{1}, α_{2} \geq 0$ , then there exist $j \neq i$ such that $d_{i} | d_{j}$ . We show that this result remains true if $d_{i} = p_{1}^{α_{1}} \dots p_{n}^{α_{n}}$ for a fixed set ${p_{1}, \dots, p_{n}}$ of $n$ prime numbers.

Soit ${d_{i} n + b_{i} : n \in ℤ}_{i \in I}$ une famille de suites arithmétiques qui est une couverture disjointe de l’ensemble des nombres entiers. Barát and Varjú [1] ont prouvé que si $d_{i} = p_{1}^{α_{1}} p_{2}^{α_{2}}$ pour deux nombres premiers $p_{1}$ , $p_{2}$ et des entiers $α_{1}, α_{2} \geq 0$ , alors il existe $i$ et $j$ tels que $j \neq i$ et $d_{i} | d_{j}$ . Nous montrons que ce résultat reste vrai si $d_{i} = p_{1}^{α_{1}} \dots p_{n}^{α_{n}}$ pour un ensemble fixé ${p_{1}, \dots, p_{n}}$ de $n$ nombres premiers.

Received: 2020-11-28
Accepted: 2021-04-19
Published online: 2022-10-24

DOI: 10.5802/jtnb.1212

Classification: 11B25
Keywords: Arithmetic progression, covering

Author's affiliations:

Bollobás, Béla ^{1, 2}

¹ Department of Pure Mathematics and Mathematical Statistics, Wilberforce Road, Cambridge, CB3 0WA, UK
² Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152, USA

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Bollobás, Béla. Remarks on a paper of J. Barát and P.P. Varjú. Journal de théorie des nombres de Bordeaux, Volume 34 (2022) no. 2, pp. 515-516. doi : 10.5802/jtnb.1212. http://www.numdam.org/articles/10.5802/jtnb.1212/

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