# A contribution to infinite disjoint covering systems

János Barát[1]; Péter P. Varjú[1]

• [1] Bolyai Institute University of Szeged Aradi vértanúk tere 1. Szeged, 6720 Hungary
• Volume: 17, Issue: 1, page 51-55
• ISSN: 1246-7405

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## Abstract

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Let the collection of arithmetic sequences ${\left\{{d}_{i}n+{b}_{i}:n\in ℤ\right\}}_{i\in I}$ 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 $k,l\ge 0$, then there is a $j\ne i$ 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 $1254$.

## How to cite

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Barát, János, and Varjú, Péter P.. "A contribution to infinite disjoint covering systems." Journal de Théorie des Nombres de Bordeaux 17.1 (2005): 51-55. <http://eudml.org/doc/249436>.

@article{Barát2005,
abstract = {Let the collection of arithmetic sequences $\lbrace d_in+b_i:n\in \mathbb\{Z\}\rbrace _\{i\in I\}$ be a disjoint covering system of the integers. We prove that if $d_i=p^kq^l$ for some primes $p,q$ and integers $k,l\ge 0$, then there is a $j\ne i$ 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 $1254$.},
affiliation = {Bolyai Institute University of Szeged Aradi vértanúk tere 1. Szeged, 6720 Hungary; Bolyai Institute University of Szeged Aradi vértanúk tere 1. Szeged, 6720 Hungary},
author = {Barát, János, Varjú, Péter P.},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {infinite disjoint covering system; Beatty sequence; arithmetic sequence; Fraenkel's conjecture; Schinzel's conjecture},
language = {eng},
number = {1},
pages = {51-55},
publisher = {Université Bordeaux 1},
title = {A contribution to infinite disjoint covering systems},
url = {http://eudml.org/doc/249436},
volume = {17},
year = {2005},
}

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
PB - Université Bordeaux 1
VL - 17
IS - 1
SP - 51
EP - 55
AB - Let the collection of arithmetic sequences $\lbrace d_in+b_i:n\in \mathbb{Z}\rbrace _{i\in I}$ be a disjoint covering system of the integers. We prove that if $d_i=p^kq^l$ for some primes $p,q$ and integers $k,l\ge 0$, then there is a $j\ne i$ 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 $1254$.
LA - eng
KW - infinite disjoint covering system; Beatty sequence; arithmetic sequence; Fraenkel's conjecture; Schinzel's conjecture
UR - http://eudml.org/doc/249436
ER -

## References

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1. J. Barát, P.P. Varjú, Partitioning the positive integers to seven Beatty sequences. Indag. Math. 14 (2003), 149–161. Zbl1052.11018MR2026810
2. A.S. Fraenkel, Complementing and exactly covering sequences. J. Combin. Theory Ser. A 14 (1973), 8–20. Zbl0257.05023MR309770
3. A.S. Fraenkel, R.J. Simpson, On infinite disjoint covering systems. Proc. Amer. Math. Soc. 119 (1993), 5–9. Zbl0784.05011MR1148023
4. R.L. Graham, Covering the positive integers by disjoint sets of the form $\left\{\left[n\alpha +\beta \right]:n=1,2,...\right\}$. J. Combin Theory Ser. A 15 (1973), 354–358. Zbl0279.10042
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. Zbl0862.11009MR1301513
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. Zbl1055.11007MR1954716
8. Š. Porubský, Covering systems and generating functions. Acta Arithm. 26 (1975), 223–231. Zbl0268.10044MR379423
9. R.J. Simpson, Disjoint covering systems of rational Beatty sequences. Discrete Math. 92 (1991), 361–369. Zbl0742.11011MR1140599
10. S.K. Stein, Unions of Arithmetic Sequences. Math. Annalen 134 (1958), 289–294. Zbl0084.04302MR93493
11. R. Tijdeman, Fraenkel’s conjecture for six sequences. Discrete Math. 222 (2000), 223–234. Zbl1101.11314MR1771401

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