Tetration of $\pi$: Can it be a prime number? Given the tetration as
\begin{align}
{^{n}a} = \underbrace{a^{a^{\cdot^{\cdot^{a}}}}}_n
\end{align}
and the set of prime numbers as $\mathbb{P}$.
Can you prove or to disprove the following statement?
\begin{align}
\forall p\in\mathbb{P}:(^{p^2+1}\pi)\in\mathbb{P}
\end{align}
I tried to disprove it, but I failed to do it.
Thank you
 A: I believe that nothing close to this is known currently; see e.g. this answer and the ensuing discussion. There are lots of "basic" problems about integrality/nonintegrality, rationality/irrationality, algebraicity/transcendence, etc. which are wide open.
There are many number-theoretic conjectures with good heuristic evidence which are known to resolve many such questions, but I don't know of any which would resolve this particular one. I suspect that all heuristics currently known suggest that $^n\pi$ cannot even be an integer for $n\in\mathbb{N}$ (number theorists, correct me if I'm wrong about this), but I don't think we're anywhere close to proving that or any related claim. Even numerical evidence is hard to come by: $^4\pi$ is already so huge that we don't obviously have a method to check experimentally whether it "looks like" an integer.

EDIT: Adding to address questions along the lines of xakepp35's comments below, the order of exponentiation matters. For example, $$2^{(2^{(2^2)})}=2^{16}=65536$$ but $$(2^2)^{(2^2)}=4^4=256.$$ Of course the latter is much easier to calculate, and it's not hard to show that $(\pi^\pi)^{(\pi^\pi)}$ is not an integer.
Given that it's so intractable, why to people care about tetration? Well, many don't - it certainly doesn't have any real-world applications I know of, and it's a very niche topic in pure mathematics. It is however a simple example example of a very fast-growing function, and the general study of fast-growing functions (indeed, ones much faster than tetration) is quite interesting; they arise in Ramsey theory and proof theory, for example. (That first link is to Graham's number, which is unspeakably gigantic; it's definitely worth a read.)
