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 {d i n+b i :n} iI 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,l0, then there is a ji 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 {d i n+b i :n} iI 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,l0, il existe alors un ji 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.

Published online:
DOI: 10.5802/jtnb.476
János Barát 1; Péter P. Varjú 1

1 Bolyai Institute University of Szeged Aradi vértanúk tere 1. Szeged, 6720 Hungary
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János Barát; Péter P. Varjú. 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. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.476/

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