A contribution to infinite disjoint covering systems
Journal de théorie des nombres de Bordeaux, Volume 17 (2005) no. 1, pp. 51-55.

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 $1254$.

Supposons que la famille de suites arithmétiques ${\left\{{d}_{i}n+{b}_{i}:n\in ℤ\right\}}_{i\in I}$ 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 $k,l\ge 0$, il existe alors un $j\ne i$ 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 $1254$.

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

[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 $\left\{\left[n\alpha +\beta \right]:n=1,2,...\right\}$. 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

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