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February, 2022 A Generalization of Piatetski–Shapiro Sequences
Victor Zhenyu Guo, Jinyun Qi
Author Affiliations +
Taiwanese J. Math. 26(1): 33-47 (February, 2022). DOI: 10.11650/tjm/210802

Abstract

We consider a generalization of Piatetski–Shapiro sequences in the sense of Beatty sequences, which is of the form $(\lfloor \alpha n^c + \beta \rfloor)_{n=1}^{\infty}$ with real numbers $\alpha \geq 1$, $c \gt 1$ and $\beta$. We show there are infinitely many primes in the generalized Piatetski–Shapiro sequence with $c \in (1,14/13)$. Moreover, we prove there are infinitely many Carmichael numbers composed entirely of the primes from the generalized Piatetski–Shapiro sequences with $c \in (1,64/63)$.

Funding Statement

The first author is supported in part by the National Natural Science Foundation of China (No. 11901447), the China Postdoctoral Science Foundation (No. 2019M653576) and the Natural Science Foundation of Shaanxi Province (No. 2020JQ-009). The second author is supported in part by the National Natural Science Foundation of China (No. 11971381, No. 11701447, No. 11871317 and No. 11971382).

Acknowledgments

The authors thank the referees for their valuable comments. The authors also thank Prof. Yuan Yi and Prof. Yaming Lu for several helpful discussions.

Citation

Download Citation

Victor Zhenyu Guo. Jinyun Qi. "A Generalization of Piatetski–Shapiro Sequences." Taiwanese J. Math. 26 (1) 33 - 47, February, 2022. https://doi.org/10.11650/tjm/210802

Information

Received: 1 April 2021; Revised: 21 May 2021; Accepted: 11 August 2021; Published: February, 2022
First available in Project Euclid: 16 August 2021

MathSciNet: MR4367785
zbMATH: 07481701
Digital Object Identifier: 10.11650/tjm/210802

Subjects:
Primary: 11B83 , 11L07 , 11N13

Keywords: arithmetic progression , exponential sums , Piatetski–Shapiro sequences

Rights: Copyright © 2022 The Mathematical Society of the Republic of China

Vol.26 • No. 1 • February, 2022
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