For
$c\,>\,1$
we denote by
${{\pi }_{c}}\left( x \right)$
the number of integers
$n\,\le \,x$
such that
$\left\lfloor {{n}^{c}} \right\rfloor $
is prime. In 1953, Piatetski-Shapiro has proved that
${{\pi }_{c}}\left( x \right)\,\sim \,\frac{x}{c\,\log \,x},\,x\to \,+\infty $
holds for
$c\,<\,12/11$
. Many authors have extended this range, which measures our progress in exponential sums techniques. In this article we obtain
$c\,<\,1.16117\ldots $
.