Beatty primes from fractional powers of almost-primes

TL;DR

This paper proves that for certain irrational numbers and fractional powers, there are infinitely many primes within the intersection of Beatty sequences and almost-primes, extending prime distribution results in these special sequences.

Contribution

It establishes the existence of infinitely many primes in the intersection of Beatty sequences and fractional powers of almost-primes, with explicit bounds depending on the type of irrational.

Findings

Infinitely many primes in Beatty sequences intersecting fractional powers of almost-primes.

Explicit bounds on the fractional power exponent c for prime occurrence.

Extension of prime distribution results to sequences defined by irrational parameters.

Abstract

Let $\alpha>1$ be irrational and of finite type, $\beta\in\mathbb{R}$. In this paper, it is proved that for $R\geqslant13$ and any fixed $c\in(1,c_R)$, there exist infinitely many primes in the intersection of Beatty sequence $\mathcal{B}_{\alpha,\beta}$ and $\lfloor n^c\rfloor$, where $c_R$ is an explicit constant depending on $R$ herein, $n$ is a natural number with at most $R$ prime factors, counted with multiplicity.