Prime numbers in a sparse set

Yes, there is a $c > 1$ for which infinitely many numbers of the form $\lfloor k^{c} \rfloor$ are prime. The first result of this type was proven in Ilya Piatetski-Shapiro's Ph.D. thesis (written in 1954 under the direction of Alexander Buchstab) and holds for any $1 \leq c \leq 12/11$. A reference (from Wikipedia) is Pyateckiĭ-Šapiro, I.I. (1953). "On the distribution of prime numbers in sequences of the form [f(n)]". Mat. Sbornik N.S. 33 (75): 559–566.

The largest currently know range of $c$ values is $1 \leq c < 243/205$ due to Rivat and Wu. (Glasgow Math Journal, 2001, volume 43, no. 2, 237-254.)

I should mention that aside from primes represented by the polynomial $x^2 + y^4$, we may further thin the sequence by insisting that $y$ is also prime; this is a result of Heath-Brown and Li. Moreover, in 2001 Heath-Brown showed that the binary cubic form $x^3 + 2y^3$ represents infinitely many primes, thereby resolving an old question of Hardy which asked whether there are infinitely many primes which are sums of three cubes. Heath-Brown's result was subsequently generalized in joint work with B. Moroz in 2002.