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A contribution to infinite disjoint covering systems

János Barát;  Péter P. Varjú — 2005

Journal de Théorie des Nombres de Bordeaux

Let the collection of arithmetic sequences { d i n + b i : n ∈ ℤ } i ∈ 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 ≥ 0 , then there is a j ≠ 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 .

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