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)?



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



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> 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.
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This is only a naive question, thanks in advance.



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.)


