| 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. |
