Prime numbers in a sparse set Is there any set $X$ which is a density 0 subset of $N^*$ and we already know that there are infinitely many primes in it, beside examples which come from $x^2+y^4$(or its proof)?

Problem1: In particular, is it already proved that there exist $c>1$, s.t. $A_c=\{n\in\mathbb{N}^*| \exists k\in\mathbb{N}^* , n=[k^c]\}$ contains infinitely many primes?

If this problem can not be solved by existing methods, can the following one be solved by existing methods?

Problem2: for all $c>1$, $\exists d(c), c>d(c)>0$, s.t. $A_c=\{n\in\mathbb{N}^*| \exists k\in\mathbb{N}^* , n=[k^{e}], e\in (c-d(c),c+d(c))\}$ contains infinitely many primes.

This is only a naive question, thanks in advance.
 A: 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.
A: 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.)
