I'm looking for a result (embarrassingly enough, a somewhat famous result) which shows the infinitude in some sense I don't recall of primes of the form $$ \lfloor x^k\rfloor $$ for $k$ fixed and irrational. There were sharp limits on the size of $k.$ I think the original result has been improved many times, mostly by widening the allowable range of $k.$
Rivat and Sargos, Nombres premiers de la forme $[n^c]$, Canad. J. Math. 53 (2001), no. 2, 414–433, MR1820915 (2002a:11107), reviewed by G. Greaves.
The authors establish an asymptotic formula for the number of primes not exceeding $x$ of the form $[n^c]$. Their result applies for each $c$ with $1\lt c\lt2817/2426$. The review compares this to previous work, and there are links to other papers and reviews that cite this paper.
Apparently the first paper along these lines was by Piatetski-Shapiro in 1953, with $1\lt c\lt12/11$.