A generalization of Piatetski-Shapiro sequences (II)

Abstract

Suppose that $\alpha ,\beta \in \mathbb{R}$. Let $\alpha ⩾1$ and $c$ be a real number in the range $1<c<12/11$. In this paper, it is proved that there exist infinitely many primes in the generalized Piatetski--Shapiro sequence, which is defined by $(\lfloor \alpha n^c+\beta \rfloor {)}_{n=1}^{\infty }$. Moreover, we also prove that there exist infinitely many Carmichael numbers composed entirely of primes from the generalized Piatetski--Shapiro sequences with $c\in (1,\frac{19137}{18746})$. The two theorems constitute improvements upon previous results by Guo and Qi.

12 pages. arXiv admin note: text overlap with arXiv:2109.00461