The author proves that, if \(1<c<20/17\), then the number of primes having the form \([n^ c]\) with \(n\leq x\) is \(\gg x/\log x\). He has subsequently improved this to \(1<c<13/11\) in his second paper [Sci. China, Ser. A 36, No. 8, 913-926 (1993; Zbl 0790.11063)], which has already been reviewed.