# 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

Journal de Théorie des Nombres de Bordeaux (2005)

- Volume: 17, Issue: 1, page 51-55
- ISSN: 1246-7405

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topBarát, 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

top- J. Barát, P.P. Varjú, Partitioning the positive integers to seven Beatty sequences. Indag. Math. 14 (2003), 149–161. Zbl1052.11018MR2026810
- A.S. Fraenkel, Complementing and exactly covering sequences. J. Combin. Theory Ser. A 14 (1973), 8–20. Zbl0257.05023MR309770
- A.S. Fraenkel, R.J. Simpson, On infinite disjoint covering systems. Proc. Amer. Math. Soc. 119 (1993), 5–9. Zbl0784.05011MR1148023
- R.L. Graham, Covering the positive integers by disjoint sets of the form $\left\{\right[n\alpha +\beta ]:n=1,2,...\}$. J. Combin Theory Ser. A 15 (1973), 354–358. Zbl0279.10042
- C.E. Krukenberg, Covering sets of the integers. Ph. D. Thesis, Univ. of Illinois, Urbana-Champaign, IL, (1971)
- E. Lewis, Infinite covering systems of congruences which don’t exist. Proc. Amer. Math. Soc. 124 (1996), 355–360. Zbl0862.11009MR1301513
- 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
- Š. Porubský, Covering systems and generating functions. Acta Arithm. 26 (1975), 223–231. Zbl0268.10044MR379423
- R.J. Simpson, Disjoint covering systems of rational Beatty sequences. Discrete Math. 92 (1991), 361–369. Zbl0742.11011MR1140599
- S.K. Stein, Unions of Arithmetic Sequences. Math. Annalen 134 (1958), 289–294. Zbl0084.04302MR93493
- R. Tijdeman, Fraenkel’s conjecture for six sequences. Discrete Math. 222 (2000), 223–234. Zbl1101.11314MR1771401

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