Let the collection of arithmetic sequences be a disjoint covering system of the integers. We prove that if for some primes and integers , then there is a such that . 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 . The above conjecture holds for saturated systems with such that the product of its prime factors is at most .
Supposons que la famille de suites arithmétiques soit un recouvrement disjoint des nombres entiers. Nous prouvons qui si pour des nombres premiers et des entiers , il existe alors un tel que . On conjecture que le résultat de divisibilité est vrai quelques soient les raisons .
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 dont le produit des facteurs premiers n’est pas supérieur à .
@article{JTNB_2005__17_1_51_0, author = {Bar\'at, J\'anos and Varj\'u, P\'eter P.}, 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 = {http://www.numdam.org/articles/10.5802/jtnb.476/} }
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 SP - 51 EP - 55 VL - 17 IS - 1 PB - Université Bordeaux 1 UR - http://www.numdam.org/articles/10.5802/jtnb.476/ DO - 10.5802/jtnb.476 LA - en ID - JTNB_2005__17_1_51_0 ER -
%0 Journal Article %A Barát, János %A Varjú, Péter P. %T A contribution to infinite disjoint covering systems %J Journal de théorie des nombres de Bordeaux %D 2005 %P 51-55 %V 17 %N 1 %I Université Bordeaux 1 %U http://www.numdam.org/articles/10.5802/jtnb.476/ %R 10.5802/jtnb.476 %G en %F JTNB_2005__17_1_51_0
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/
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