Abstract
We prove that the sequence [ξ(5/4)n], n=1,2, . . . , where ξ is an arbitrary positive number, contains infinitely many composite numbers. A corresponding result for the sequences [(3/2)n] and [(4/3)n],n=1,2, . . . , was obtained by Forman and Shapiro in 1967. Furthermore, it is shown that there are infinitely many positive integers n such that ([ξ(5/4)n],6006)>1, where 6006=2·3·7·11·13. Similar results are obtained for shifted powers of some other rational numbers. In particular, the same is proved for the sets of integers nearest to ξ(5/3)n and to ξ(7/5)n, n∈ℕ. The corresponding sets of possible divisors are also described.
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Dubickas, A., Novikas, A. Integer parts of powers of rational numbers. Math. Z. 251, 635–648 (2005). https://doi.org/10.1007/s00209-005-0827-4
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DOI: https://doi.org/10.1007/s00209-005-0827-4