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.

Let ${\left\{{d}_{i}n+{b}_{i}:\phantom{\rule{4pt}{0ex}}n\in ℤ\right\}}_{i\in I}$ be a family of disjoint arithmetic progressions covering the integers. Barát and Varjú [1] have proved that if ${d}_{i}={p}_{1}^{{\alpha }_{1}}{p}_{2}^{{\alpha }_{2}}$ for two prime numbers ${p}_{1}$, ${p}_{2}$ and integers des ${\alpha }_{1},{\alpha }_{2}\ge 0$, then there exist $j\ne i$ such that ${d}_{i}|{d}_{j}$. We show that this result remains true if ${d}_{i}={p}_{1}^{{\alpha }_{1}}\cdots {p}_{n}^{{\alpha }_{n}}$ for a fixed set $\left\{{p}_{1},\cdots ,{p}_{n}\right\}$ of $n$ prime numbers.

Soit ${\left\{{d}_{i}n+{b}_{i}:\phantom{\rule{4pt}{0ex}}n\in ℤ\right\}}_{i\in I}$ une famille de suites arithmétiques qui est une couverture disjointe de l’ensemble des nombres entiers. Barát and Varjú [1] ont prouvé que si ${d}_{i}={p}_{1}^{{\alpha }_{1}}{p}_{2}^{{\alpha }_{2}}$ pour deux nombres premiers ${p}_{1}$, ${p}_{2}$ et des entiers ${\alpha }_{1},{\alpha }_{2}\ge 0$, alors il existe $i$ et $j$ tels que $j\ne i$ et ${d}_{i}|{d}_{j}$. Nous montrons que ce résultat reste vrai si ${d}_{i}={p}_{1}^{{\alpha }_{1}}\cdots {p}_{n}^{{\alpha }_{n}}$ pour un ensemble fixé $\left\{{p}_{1},\cdots ,{p}_{n}\right\}$ de $n$ nombres premiers.

Accepted:
Published online:
DOI: 10.5802/jtnb.1212
Classification: 11B25
Keywords: Arithmetic progression, covering
Béla Bollobás 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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Béla Bollobás. 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. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1212/

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