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 {d i n+b i :n} iI 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 0, then there exist ji such that d i |d j . We show that this result remains true if d i =p 1 α 1 p n α n for a fixed set {p 1 ,,p n } of n prime numbers.

Soit {d i n+b i :n} iI 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 0, alors il existe i et j tels que ji et d i |d j . Nous montrons que ce résultat reste vrai si d i =p 1 α 1 p n α n pour un ensemble fixé {p 1 ,,p n } de n nombres premiers.

Received:
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
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
@article{JTNB_2022__34_2_515_0,
     author = {B\'ela Bollob\'as},
     title = {Remarks on a paper of {J.~Bar\'at} and {P.P.~Varj\'u}},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {515--516},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {34},
     number = {2},
     year = {2022},
     doi = {10.5802/jtnb.1212},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1212/}
}
TY  - JOUR
AU  - Béla Bollobás
TI  - Remarks on a paper of J. Barát and P.P. Varjú
JO  - Journal de théorie des nombres de Bordeaux
PY  - 2022
SP  - 515
EP  - 516
VL  - 34
IS  - 2
PB  - Société Arithmétique de Bordeaux
UR  - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1212/
DO  - 10.5802/jtnb.1212
LA  - en
ID  - JTNB_2022__34_2_515_0
ER  - 
%0 Journal Article
%A Béla Bollobás
%T Remarks on a paper of J. Barát and P.P. Varjú
%J Journal de théorie des nombres de Bordeaux
%D 2022
%P 515-516
%V 34
%N 2
%I Société Arithmétique de Bordeaux
%U https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1212/
%R 10.5802/jtnb.1212
%G en
%F JTNB_2022__34_2_515_0
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/

[1] János Barát; Péter P. Varjú A contribution to infinite disjoint covering systems, J. Théor. Nombres Bordeaux, Volume 17 (2005) no. 1, pp. 51-55 | DOI | Numdam | MR | Zbl

[2] Leonard E. Dickson Finiteness of the odd perfect and primitive abundant numbers with n distinct prime factors, Am. J. Math., Volume 35 (1913), pp. 413-422 | DOI | MR | Zbl

[3] Pál Erdős On integers of the form 2 k +p and some related problems, Summa Brasil. Math., Volume 2 (1950), pp. 113-123 | MR | Zbl

[4] Pál Erdős Quelques problèmes de théorie des nombres, Introduction à la théorie des nombres (Monographies de l’Enseignement Mathématique), Volume 6, L’Enseignement Mathématique, 1963, pp. 81-135 | Zbl

[5] Joseph B. Kruskal The theory of well-quasi-ordering: A frequently discovered concept, J. Comb. Theory, Ser. A, Volume 13 (1972), pp. 297-305 | DOI | MR | Zbl

[6] Neil Robertson; Paul D. Seymour Graph minors XIX, Well-quasi-ordering on a surface, J. Comb. Theory, Ser. B, Volume 90 (2004) no. 2, pp. 325-385 | DOI | MR | Zbl

[7] Neil Robertson; Paul D. Seymour Graph minors XX, Wagner’s conjecture, J. Comb. Theory, Ser. B, Volume 92 (2004) no. 2, pp. 325-357 | DOI | MR | Zbl

Cited by Sources: