A contribution to infinite disjoint covering systems
Journal de Théorie des Nombres de Bordeaux, Tome 17 (2005) no. 1, pp. 51-55.

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.

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.

@article{JTNB_2005__17_1_51_0,
     author = {J\'anos Bar\'at and P\'eter P. Varj\'u},
     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},
     zbl = {1079.11008},
     mrnumber = {2152210},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.476/}
}
János Barát; Péter P. Varjú. 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. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.476/

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