Tetration and p-adic

[marcokrt]

EDIT:!!!!!

Okay, so I can't prove this. But I believe Marco has proven that:

\[
\lim_{N\to \infty}\,\,^N a = A \in \mathbb{Q}_p\\
\]

A (somewhat provocative) simple question: Evaluate A of

\[
a=0\in\mathbb{Q}_2, \lim_{N\to \infty}\,\,^N a = A \in \mathbb{Q}_2\\
\]

What is A? 0 or 1? and \( \mathbb{Q}_3,\mathbb{Q}_5,\mathbb{Q}_7  \)?

By Definition 1.3 from the mentioned reference "Number of stable digits of any integer tetration" (https://arxiv.org/pdf/2210.07956.pdf), we have that \( V(0):=0 \), since \(^N 0 = 0 \) iff \( N \) is odd and \(^N 0 = 1 \) otherwise. Now, I introduced this definition in the paper, since we want that the "constant congruence speed" of tetration is a function from the nonnegative integers which are not a multiple of \( 10 \) and the whole set \( \mathbb{N}_0 \), but this is not the unique possible solution of the underlined issue. Here is the most interesting point: we may assume that \( V(0) = \infty \) by looking at the solution \( \alpha_{00} \) of the fundamental equation \(y^5=y \) in the commutative ring \( \mathbb{Z}_{10} \) (see page 11, ibid.), since this is the only solution linked to the constant congruence speed of the tetration bases that are congruent to \( 0 \) modulo \( 10 \), and thus its p-adic valuation would always be \( +\infty \). Thus, we would have that \( V(0) = +\infty \) also in different numerical systems, if we assume that the first line of Equation 16 is given by \( v_p(0) \), where \( v_p \) indicates the p-adic valuation of the argument for any given prime number \( p \).
So, the best answer to your question above cannot ignore our preliminary assumption based on the fact that \( V(0) \in \mathbb{Z} \) definition, while it would be not even be an element \( \mathbb{R} \ by construction... otherwise the "constant congruence speed analysis" cannot be taken as a way to solve related problems in the p-adics.