On Piatetski-Shapiro primes from almost primes

Abstract

Denote by ${\mathcal{P}}_r$ an almost-prime with at most $r$ prime factors, counted according to multiplicity. In this manuscript, it is established that, for any fixed $0.98353<\gamma <1$, there exist infinitely many primes of the form $p=[n^{1/\gamma }]$, where $n$ is an almost-prime ${\mathcal{P}}_7$. This result constitutes an improvement upon the previous result of Baker, Banks, Guo and Yeager [1], who showed that there exist infinitely many primes $p$ such that $p=[n^{1/\gamma }]$ with $n\in {\mathcal{P}}_8$ for $\gamma$ near to one.

20 pages