A contribution to infinite disjoint covering systems

Barát, János; Varjú, Péter P.

doi:10.5802/jtnb.476

Barát, János ¹ ; Varjú, Péter P.¹

¹ Bolyai Institute University of Szeged Aradi vértanúk tere 1. Szeged, 6720 Hungary

Journal de théorie des nombres de Bordeaux, Tome 17 (2005) no. 1, pp. 51-55.

Résumé
Abstract

Supposons que la famille de suites arithmétiques ${d_{i} n + b_{i} : n \in ℤ}_{i \in I}$ soit un recouvrement disjoint des nombres entiers. Nous prouvons qui si $d_{i} = p^{k} q^{l}$ pour des nombres premiers $p, q$ et des entiers $k, l \geq 0$ , il existe alors un $j \neq i$ tel que $d_{i} | d_{j}$ . On conjecture que le résultat de divisibilité est vrai quelques soient les raisons $d_{i}$ .

Un recouvrement disjoint est appelé saturé si la somme des inverses des raisons est égale à 1. La conjecture ci-dessus est vraie pour des recouvrements saturés avec des $d_{i}$ dont le produit des facteurs premiers n’est pas supérieur à $1254$ .

Let the collection of arithmetic sequences ${d_{i} n + b_{i} : n \in ℤ}_{i \in I}$ be a disjoint covering system of the integers. We prove that if $d_{i} = p^{k} q^{l}$ for some primes $p, q$ and integers $k, l \geq 0$ , then there is a $j \neq i$ such that $d_{i} | d_{j}$ . We conjecture that the divisibility result holds for all moduli.

A disjoint covering system is called saturated if the sum of the reciprocals of the moduli is equal to $1$ . The above conjecture holds for saturated systems with $d_{i}$ such that the product of its prime factors is at most $1254$ .

EuDML MR Zbl | 1 citation dans Numdam

DOI : 10.5802/jtnb.476

Affiliations des auteurs :

Barát, János ¹ ; Varjú, Péter P. ¹

¹ Bolyai Institute University of Szeged Aradi vértanúk tere 1. Szeged, 6720 Hungary

@article{JTNB_2005__17_1_51_0,
     author = {Bar\'at, J\'anos and Varj\'u, P\'eter P.},
     title = {A contribution to infinite disjoint covering systems},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {51--55},
     publisher = {Universit\'e Bordeaux 1},
     volume = {17},
     number = {1},
     year = {2005},
     doi = {10.5802/jtnb.476},
     zbl = {1079.11008},
     mrnumber = {2152210},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/jtnb.476/}
}

TY  - JOUR
AU  - Barát, János
AU  - Varjú, Péter P.
TI  - A contribution to infinite disjoint covering systems
JO  - Journal de théorie des nombres de Bordeaux
PY  - 2005
SP  - 51
EP  - 55
VL  - 17
IS  - 1
PB  - Université Bordeaux 1
UR  - http://www.numdam.org/articles/10.5802/jtnb.476/
DO  - 10.5802/jtnb.476
LA  - en
ID  - JTNB_2005__17_1_51_0
ER  -

%0 Journal Article
%A Barát, János
%A Varjú, Péter P.
%T A contribution to infinite disjoint covering systems
%J Journal de théorie des nombres de Bordeaux
%D 2005
%P 51-55
%V 17
%N 1
%I Université Bordeaux 1
%U http://www.numdam.org/articles/10.5802/jtnb.476/
%R 10.5802/jtnb.476
%G en
%F JTNB_2005__17_1_51_0

Barát, János; Varjú, Péter P. A contribution to infinite disjoint covering systems. Journal de théorie des nombres de Bordeaux, Tome 17 (2005) no. 1, pp. 51-55. doi : 10.5802/jtnb.476. http://www.numdam.org/articles/10.5802/jtnb.476/

Bibliographie
Cité par

[1] J. Barát, P.P. Varjú, Partitioning the positive integers to seven Beatty sequences. Indag. Math. 14 (2003), 149–161. | MR | Zbl

[2] A.S. Fraenkel, Complementing and exactly covering sequences. J. Combin. Theory Ser. A 14 (1973), 8–20. | MR | Zbl

[3] A.S. Fraenkel, R.J. Simpson, On infinite disjoint covering systems. Proc. Amer. Math. Soc. 119 (1993), 5–9. | MR | Zbl

[4] R.L. Graham, Covering the positive integers by disjoint sets of the form ${[n α + β] : n = 1, 2, ...}$ . J. Combin Theory Ser. A 15 (1973), 354–358. | Zbl

[5] C.E. Krukenberg, Covering sets of the integers. Ph. D. Thesis, Univ. of Illinois, Urbana-Champaign, IL, (1971)

[6] E. Lewis, Infinite covering systems of congruences which don’t exist. Proc. Amer. Math. Soc. 124 (1996), 355–360. | MR | Zbl

[7] Š. Porubský, J. Schönheim, Covering Systems of Paul Erdős, Past, Present and Future. In Paul Erdős and his Mathematics I., Springer, Budapest, (2002), 581–627. | MR | Zbl

[8] Š. Porubský, Covering systems and generating functions. Acta Arithm. 26 (1975), 223–231. | MR | Zbl

[9] R.J. Simpson, Disjoint covering systems of rational Beatty sequences. Discrete Math. 92 (1991), 361–369. | MR | Zbl

[10] S.K. Stein, Unions of Arithmetic Sequences. Math. Annalen 134 (1958), 289–294. | MR | Zbl

[11] R. Tijdeman, Fraenkel’s conjecture for six sequences. Discrete Math. 222 (2000), 223–234. | MR | Zbl

Cité par Sources :