Piatetski-Shapiro Primes in a Beatty Sequence

TL;DR

This paper proves the existence of infinitely many Piatetski-Shapiro primes within a non-homogeneous Beatty sequence for certain irrational parameters and a specific range of the exponent c.

Contribution

It establishes the infinitude of Piatetski-Shapiro primes in non-homogeneous Beatty sequences under new conditions involving irrationality and finite type.

Findings

Infinitely many Piatetski-Shapiro primes in the sequence for 1<c<14/13.

Conditions on α and β ensure primes are found in the Beatty sequence.

Extension of prime distribution results to non-homogeneous Beatty sequences.

Abstract

Let $\alpha,\beta$ be real numbers such that $\alpha>1$ is irrational and of finite type, and let $c$ be a real number in the range $1<c<\frac{14}{13}$. In this paper, it is shown that there are infinitely many Piatetski-Shapiro primes $p = \left\lfloor n^c \right\rfloor$ in the non-homogenous Beatty sequence $\big(\left\lfloor\alpha m+\beta\right\rfloor\big)_{m=1}^\infty$.