TITLE: Prime numbers in a sparse set
QUESTION [7 upvotes]: 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.

REPLY [6 votes]: 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.