$\newcommand\tetra[2]{{^{#1}{#2}}}$In a recent discussion on the Tetration Forum (see <https://math.eretrandre.org/tetrationforum/showthread.php?tid=1703&page=2>), it has been pointed out how my results on the constancy of the "congruence speed" of the integer tetration $\tetra b a$ (a peculiar property of hyper-$4$ described in [The congruence speed formula](https://arxiv.org/abs/2208.02622)), would imply that $\lim\_{b \rightarrow \infty} \tetra b a$ is always an element of the set $\mathbb{Q}\_p$.


At the time, I did not fully realize the power of this breakthrough, since my original goal was only to find a function $V(a) : \mathbb{Z}^+-\{M\} \rightarrow \mathbb{Z}^+$, where $M = \{a : a \not\equiv 0 \pmod {10} \land a \neq 1\}$, but after a little search, I have seen that this is a general open problem, explicitly solved only for a few cases (since 1953, for $a:=2$). Furthermore, I have not managed to find any proof that even $\lim\_{b \rightarrow \infty} \tetra b 6$ is a rational $p$-adic number (or rather an irrational one, disproving the aforementioned claim).


Now, my first concern is if I am wrong and $\lim\_{b \rightarrow \infty} \tetra b 6$ has already been proven to be an element of $\mathbb{Q}\_p$ (or not), while my general request here is to know more about the implications of Equation 16 on page 15 of [Ripà and Onnis - Number of stable digits of any integer tetration](https://arxiv.org/abs/2210.07956) on the above mentioned general open problem (and related topics).



$\lim_{b \rightarrow \infty} {^{b}a} \in \mathbb{Q}_p$ for any $a \in \mathbb{Z}^+$?
A sequence given by $x\_1=a$, $x\_{n+1}=a^{x\_n}$, where $a$ is a positive integer, is eventually constant modulo every positive integer $T$. This is widely known.


A short proof. We induct in $T$, so assume that $T>1$ and the claim is proven for smaller numbers. Denote $T=T\_1T\_2$, where $T\_1$ has prime divisors which divide $a$, and $T\_2$ is coprime with $a$. Modulo $T\_1$, obviously $x\_n$ become equal to 0. Modulo $T\_2$, the value of $x\_n$ is determined by the remainder of $x\_{n-1}$ modulo $\varphi(T\_2)<T\_2$, thus by induction hypothesis it is eventually constant.


I hope that this is what was asked.


