Some arithmetic properties of numbers of the form $\lfloor p^c\rfloor$

Abstract

Let $${\mathbb{P}}^c=(\lfloor p^c\rfloor {)}_{p\in \mathbb{P}}(c>1,c\notin \mathbb{N}),$$ where $\mathbb{P}$ is the set of prime numbers, and $\lfloor \cdot \rfloor$ is the floor function. We show that for every such $c$ there are infinitely many members of ${\mathbb{P}}^c$ having at most $R(c)$ prime factors, giving explicit estimates for $R(c)$ when $c$ is near one and also when $c$ is large.

arXiv comment15 pages; the first theorem in version one of this manuscript has been removed, as the result is a special case of a more general theorem established earlier by Cao and Zhai