A finite set of prime numbers S is called unavoidable with respect to b > 1 if for each ξ ∈ ℝ the sequence of integer parts ⌊ξbn⌋, n = 0, 1, 2,…, contains infinitely many elements divisible by at least one prime number p from the set S. It is known that an unavoidable set exists with respect to b = 2, 3, 4, 6 and that it does not exist if b > 1 is an integer such that b − 1 is not square free. In this paper, we show that no finite unavoidable sets exist with respect to b if b > 3 is a prime number or b belongs to some explicitly given arithmetic progressions, for instance, 24k + 11 and 24k + 19, k = 0, 1, 2,….

中文翻译：

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从截断的周期序列获得的整数的整除性

一组有限的素数小号被称为不可避免的b > 1如果对于每个ξ ∈ ℝ整数部分的序列⌊ξbn⌋,n = 0, 1, 2,…,包含无限多个可被至少一个素数整除的元素p从集合小号. 已知存在一个不可避免的集合b = 2, 3, 4, 6并且它不存在，如果b > 1是一个整数，使得b - 1不是正方形的。在本文中，我们证明不存在关于b如果b > 3是质数或b属于一些明确给定的算术级数，例如，24ķ + 11和24ķ + 19,ķ = 0, 1, 2,….