Almost primes of the form $[p^{1/\gamma}]$

TL;DR

This paper proves that for certain values of gamma close to one, infinitely many primes p satisfy that the integer part of p^{1/γ} is an almost-prime with at most 7 prime factors, improving previous bounds.

Contribution

It improves the known bound from 8 to 7 prime factors for the almost-prime condition of the integer part of p^{1/γ} for gamma between 0.989 and 1.

Findings

Infinitely many primes p satisfy [p^{1/γ}] = P_7 for 0.989<γ<1.

Extension of previous results from P_8 to P_7.

Refinement of bounds on the prime factors of almost-primes related to p^{1/γ}.

Abstract

Let $\mathcal{P}_r$ denote an almost-prime with at most $r$ prime factors, counted according to multiplicity. In this paper, it is proved that, for $0.989<\gamma<1$, there exist infinitely many primes $p$ such that $[p^{1/\gamma}]=\mathcal{P}_7$, which constitutes an improvement upon the previous result of Banks-Guo-Shparlinski [4] who showed that there exist infinitely many primes $p$ such that $[p^{1/\gamma}]=\mathcal{P}_8$ for $\gamma$ near to one.