Anyway, thank you very much, BR and Andres, for helpful comments; I'm wondering what more can be asked or said about $\lim_{k \rightarrow \infty}2\uparrow\uparrow k \bmod N$ , since this limit seems to exist. It's a residue in $(\mathbb{Z}/N\mathbb{Z})^{*}$, and I'm now wondering about the fractions $\frac{\lim_{k \rightarrow \infty}2\uparrow\uparrow k \bmod N}{N}$.

I'm using the one which is called the Ackermann-P\'{e}ter function on Wikipedia (en.wikipedia.org/wiki/Ackermann_function). Thanks.

What about if this is tried, say, with chess variants or variants of chess pieces.

The second one: $$L\left(\left(\frac{\cdot}{q}\right),1\right) = \frac{\pi}{q^{1/2}\left(r-\left(\frac{r}{q}\right)\right)}\sum_{0<m<q/2}\left(\frac{m}{q}\right)\left(r-1-2\left\lfloor\frac{mr}{q}\right\rfloor\right)$$ and the third one: $$ L\left(\left(\frac{\cdot}{q}\right),1\right) = \frac{2\pi}{q^{1/2}\left(3-\left(\frac{3}{q}\right)\right)}\sum_{0<m<q/3}\left(\frac{m}{q}\right)$$

I can't seem to make the first three equations display properly. Here is the first one: $$L\left(\left(\frac{\cdot}{q}\right),1\right) = \frac{\pi}{q^{1/2}\left(2-\left(\frac{2}{q}\right)\right)}\sum_{0<m<q/2}\left(\frac{m}{q}\right)$$

Jeff, You can use Bateman-Horn heuristics to estimate $\left|\{f(p): p \leq x, f(p) \mbox{ is prime}\}\right|$. Use Bateman-Horn for two polynomials, set $f_1(n)=n$ and $f_2(n)=f(n)$, the cubic polynomial you want. Bateman-Horn will count the inputs $n \leq x$ for which both are prime.

Thank you very much, Professor Green. It is really interesting that the infimum goes to zero as $x \rightarrow \infty$ for the case you described! I'll look at the paper. Thanks.