The well-known Piatetski–Shapiro primes are primes of the form \([n^c]\) . In this paper, we prove that there are infinitely many Piatetski–Shapiro primes in arithmetic progressions for \(1<c<\frac{8}{7}\) . Moreover, we also prove that there exist infinitely many Carmichael numbers composed entirely of Piatetski–Shapiro primes for \(1<c<\frac{2101}{2000}\) . These two theorems constitute improvements upon the previous results.
