Tetration and p-adic

[marcokrt]

But, I fully believe that for natural numbers, Marco has shown that \(^\infty a\) for \(a \in \mathbb{N}\) is always in \(\mathbb{Q}_p\)--for every prime. And it is never irrational.

I think irrationality will happen for \(^N a\) and \(\mathbb{N}\); and \(N\)'s relation to the prime \(p\). But, these are still in \(\mathbb{Q}_p\). And I believe that Marco's result; if not proves this result, casts a wide net of results in which \(^\infty a \in \mathbb{Q}_p\). Rather than being irrational, in \(^\infty a \in \mathbb{R}_p / \mathbb{Q}_p\)...... The digits where we take Rational numbers, and tend to infinity, we see a repeatable pattern. Which looks like Marco's modular stuff. And studying that; talks about lists of digits, their repeating patterns, under tetration, under the norm where growth shrinks, and shrinking grows....

Wow, it does feel like becoming an eyewitness to a new frontier. 

My thought, exactly! There are so many things that you can do and my trilogy of papers is just a starting point looking to a new world made of discoveries and fascinating relations, not only involving the number of "convergent" digits peculiar of the integer tetration, but also the figures to their left... just like a wave in the ocean that propagates itself in the shape of concentric circles, gradually becoming more and more indistinguishable from the rest of the water (and finally leading us to some kind of caos theory approach, for the most significant digits, maybe?).
Now, just think to extend the aforementioned big world to a new dimension, by considering the reals or maybe taking into account the complex plane for a generic tetration base... we can get some surprising answers over there and I can only wish "Good luck to you all!".

I agree entirely. My field of research doesn't deal with p-adic analysis. But I have a fairly strong grasp of some advanced topics of p-adic analysis. I made it a personal goal to understand Tate's thesis--which I do to my own satisfaction. Sadly, I don't think I'd be able to produce novel results in p-adic analysis--but I can point out some things which "should happen." I'd love a hard rigor proof that \(a \in \mathbb{N}\) then \(\lim_{|N|_p \to 0}\,\,^N a \in \mathbb{Q}_p\). Which, I'm pretty sure you've shown, Marco. But I could be mistaken on some technicalities. But the essential result you showed is that:

\[
\lim_{N\to\infty}\,\, ^N a = \overline{a_1a_2\cdots a_N}A\\
\]

Where we repeat to the left; and that means the value is a rational number in p-adic circles; the same way: \(0.\overline{9} = 1\) is a rational number. I'm not the best versed in this shit though; I just know enough to get by if someone starts talking about it . Definitely continue your research though! And you're only doing favours for yourself if you phrase it in p-adic terms

You'd definitely get some eyes on your search for tenure if you proved hands down \(^\infty a \in \mathbb{Q}_p\) for all \(a \in \mathbb{N}\) and \(p\) prime--or even \(a \in \mathbb{Q}\). That's a solid, sexy, result that tenure boards love

EDIT:

Because I can't help myself. Marco has shown that:

\[
f_a(z) : \mathbb{D}_p \to \mathbb{C}_p\\
\]

Where:

\[
f_a(z) = \sum_{N=0}^\infty \,^N a z^n\\
\]

Converges for \(a \in \mathbb{N}\) and \(\mathbb{D}_p\) the p-adic unit disk. Or at least something close to this . There's definitely more finesse involved. But the p-adic field and p-adic analysis is basically normal analysis; just big means small, and small means big, lmao. There may be some more finesse with the complex plane; because \(\mathbb{C}_p\) is pretty fucked, lmao. But the correlation, would be that:

\[
F(z) = \sum_{n=0}^\infty q_k z^k\\
\]

Where \(q_k \in \mathbb{Q}\) and \(q_k \in [0,1]\)--which are a subset of entire functions. So that Marco's family of functions; is of the same type as this family.

Unfortunately I have just a degree in Economics, and I am currently focused on graph theory, so I cannot improve your analysis... I can only shortly recap here the idea shown by the last couple of papers (for a full proof of Equation 16 in https://arxiv.org/pdf/2210.07956.pdf, just follow the path "On the constant congruence speed of tetration" \( \Rightarrow \) "The congruence speed formula" and finally "Number of stable digits of any integer tetration").

We start by considering the (standard) numeral system radix-10. We know that any natural number can be represented as a 10-adic integer and we know also that the 10-adic integers form a commutative ring, so we use the well-known ring homomorphism (finding all the 15 solutions of the fundamental equation \( y^5=y \)) in order to find all the fixed points that let us apply the right period to any of them (this feature is the main achievement of "On the constant congruence speed of tetration"), returning the sets of all the natural bases of the integer tetration characterized by any given constant congruence speed.
Basically, we finally have a function between the nonnegative integers that are not a multiple of 10 and the whole set of the natural numbers, without any exception.
In the last paper, "Number of stable digits of any integer tetration" (with an inelegant proof made of very basic calculations), we start from the result above and provide the constant congruence speed of any given tetration base (as above) in terms of the valuation function (i.e., 2-adic and 5-adic valuation). This is the main result of the whole journey, achieving the first goal of "La strana coda della serie n^n^..." (2011) so that OEIS sequences as A317905 are well-defined.

Now, I think that the next step by yours/top number theorists, could be to use in the proper way more powerful tools (such as the strict p-adic analysis) in order to open the domain I've set up in the first paper of the trilogy, searching for wider/more interesting relations and going for bigger achievements. There are so many things waiting to be done around this small fixed stone put on the ground by the aforementioned trilogy of papers.