A contribution to infinite disjoint covering systems

Barát, János; Varjú, Péter P.

doi:10.5802/jtnb.476

Barát, János ; Varjú, Péter P.

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 249436 | MR 2152210 | Zbl 1079.11008

DOI : https://doi.org/10.5802/jtnb.476

BibTeX
RIS

@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},
     mrnumber = {2152210},
     zbl = {1079.11008},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/jtnb.476/}
}

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AU  - Barát, János
AU  - Varjú, Péter P.
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JO  - Journal de Théorie des Nombres de Bordeaux
PY  - 2005
DA  - 2005///
SP  - 51
EP  - 55
VL  - 17
IS  - 1
PB  - Université Bordeaux 1
UR  - http://www.numdam.org/articles/10.5802/jtnb.476/
UR  - https://www.ams.org/mathscinet-getitem?mr=2152210
UR  - https://zbmath.org/?q=an%3A1079.11008
UR  - https://doi.org/10.5802/jtnb.476
DO  - 10.5802/jtnb.476
LA  - en
ID  - JTNB_2005__17_1_51_0
ER  -

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/

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