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.

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    $\begingroup$ As stated this has trivial answers—for example, the set $X=\{$all prime numbers$\}$ has the required properties. You might want to be more specific about the types of sets you intend to consider. $\endgroup$ – Greg Martin Dec 1 '20 at 6:59

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.)

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    $\begingroup$ Do experts believe that it holds for all real $c$ from $1$ to $2$? And what is believed to be the maximal interval for which it holds? $\endgroup$ – user21820 Dec 1 '20 at 6:50
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    $\begingroup$ Yes. In a JNT article from 1983, Heath-Brown states that $1 < c < 2$ one expects $|A_{c} \cap [1,x]| \sim x/(c \log(x))$ as $x \to \infty$. Of course, there are no prime numbers of the form $\lfloor n^{2} \rfloor$, but it is natural to guess that if $c > 1$ is not an integer, then there are infinitely many primes of the form $\lfloor n^{c} \rfloor$. Indeed, Deshouillers has proven that the Lebesgue measure of the set of $c$ for which there are finitely many primes in $A_{c}$ is zero. $\endgroup$ – Jeremy Rouse Dec 1 '20 at 13:37
  • $\begingroup$ Oh okay thanks! I'm surprised that despite Deshouillers' result, we still don't know about the $n^2{+}1$ conjecture. $\endgroup$ – user21820 Dec 1 '20 at 19:50

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.


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