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

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.

arXiv comment20 pages