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arXiv · 2109.00461

Piatetski-Shapiro primes in the intersection of multiple Beatty sequences

Victor Zhenyu Guo·Jinjiang Li·Min Zhang
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Abstract

Suppose that $α_1, α_2,β_1, β_2 \in\mathbb{R}$. Let $α_1, α_2 > 1$ be irrational and of finite type such that $1, α_1^{-1}, α_2^{-1}$ are linearly independent over $\mathbb{Q}$. Let $c$ 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 intersection of Beatty sequences $\mathcal{B}_{α_1,β_1} = \lfloorα_1 n + β_1\rfloor, \mathcal{B}_{α_2, β_2} = \lfloorα_2 n + β_2\rfloor$ and the Piatetski-Shapiro sequence $\mathscr{N}^{(c)} = \lfloor n^c\rfloor$. Moreover, we also give a sketch proof of Piatetski-Shapiro primes in the intersection of multiple Beatty sequences.

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Victor Zhenyu Guo, Jinjiang Li, Min Zhang. 2021-09-01. Piatetski-Shapiro primes in the intersection of multiple Beatty sequences. https://arxiv.org/abs/2109.00461

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