Let $P(x)$ be a non-constant polynomial with real coefficients.

Can [natural density][1] of

$$\{n\ |\ \lfloor P(n)\rfloor \ \text{is prime.}\}$$

be positive? [1]: http://en.wikipedia.org/wiki/Natural_density

Let $P(x)$ be a non-constant polynomial with real coefficients.

Can natural density of

$$\{n\ |\ \lfloor P(n)\rfloor \ \text{is prime.}\}$$

be positive?

Let $P(x)$ be a non-constant polynomial with real coefficients.

Can [natural density][1] of

$$\{n\ |\ [P(n)] \ \text{is prime.}\}$$

be positive? [1]: http://en.wikipedia.org/wiki/Natural_density

Let $P(x)$ be a non-constant polynomial with real coefficients.

Can [natural density][1] of

$$\{n\ |\ \lfloor P(n)\rfloor \ \text{is prime.}\}$$

be positive? [1]: http://en.wikipedia.org/wiki/Natural_density

Let $P(x)$ be a polynomial with real coefficients.

Can [natural density][1] of

$$\{n\ |\ [P(n)] \ \text{is prime.}\}$$

be positive? [1]: http://en.wikipedia.org/wiki/Natural_density

Let $P(x)$ be a non-constant polynomial with real coefficients.

Can [natural density][1] of

$$\{n\ |\ [P(n)] \ \text{is prime.}\}$$

be positive? [1]: http://en.wikipedia.org/wiki/Natural_density