A generalization of Piatetski-Shapiro sequences (II)

TL;DR

This paper proves the existence of infinitely many primes and Carmichael numbers within a generalized Piatetski-Shapiro sequence, extending previous results and covering new ranges of the parameter c.

Contribution

It establishes the infinitude of primes and Carmichael numbers in a broader class of sequences, improving upon earlier bounds and results.

Findings

Infinitely many primes in the sequence for certain c values.

Existence of infinitely many Carmichael numbers with primes from the sequence.

Improved bounds on the parameter c for these properties.

Abstract

Suppose that $\alpha,\beta\in\mathbb{R}$. Let $\alpha\geqslant1$ 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.