Tetration and p-adic

[JmsNxn]

Read Marco's work. Marco has explained how a \(\mathbb{Z}_p\) understanding of \(^Na\) for \(a \in \mathbb{Z}_p\)--repeats as \(N\to\infty\) (through some modular pattern); where this becomes a continued fraction looking beast. And the action of \(^Na \to a^{^N a}\), the  digits repeat. And it looks like adding another repeating decimal element; just with fancier rules.

If you want to go full p-adic analysis; go full on we have some John Tate level algebraic analysis fourier magic. I don't think we're there yet. This requires deep deep insight with analysis. I was only trying to point out Marco's description as the atom blocks. Where maybe we can talk about \(\exp_p^{\circ s}(z)\) where \(z \in \mathbb{C}_p\) and \(s \in \mathbb{C}_p\) and \(p\) is a prime number.... Because we have \(^Na\) for \(a \in \mathbb{N}_p\) and \(N\to\infty\).

Gotta start with \(p\)-digit manipulations on natural numbers... That's my point. And Marco is the only person who has produced nontrivial results on this!


EDIT:!!!!!

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

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

For all \(a \in \mathbb{Q}_p\) and \(p\) is a prime. And \(N \in \mathbb{Q}_p\). But since \(\mathbb{Q}_p = \mathbb{Q}\); up until limits. We just focus on left handed repeating patterns.......

This is super weird with tetration. And 3 years ago I'd probably called this nonsense. But, fucking Marco's paper makes this "some what believable".

Lmao

I'm loaded as fuck. Did a whole party, and wanted to read tetration shit. So take my words with a grain of salt.

In \(p\)-adic Analysis, we would call Marco's result (Or at least what I believe Marco proved...):

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

If I am misinterpreting, or misrepresenting Marco's result; I can prove independently that:

\[
\lim_{|N|_p \to 0} \,\,^N a = A \in \mathbb{R}_p\\
\]

Where \(\mathbb{R}_p\) is the Cauchy closure of \(\mathbb{Q}_p\) under the measure \(|a-b|_p = p^{-\nu_p(a-b)}\).
Where the value \(\nu_p(\alpha)\) is given as:

\[
\begin{align}
\alpha &= p^{\nu_p(\alpha)} \prod_{q\neq p\,\,q \,\text{prime}} q^{\nu_q(\alpha)}\\
|\alpha|_p &= p^{-\nu_p(\alpha)}\\
0 &= p^{-\infty}
\end{align}
\]

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....

I'm definitely a little rusty on details. I know a good amount of p-adic analysis; but nothing to write home to your parents about!

[Ember Edison]

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. 

[marcokrt]

Thank you very much, dear James... yeah, I've definitely proven that stuff over the nonnegative integers, since it was the original goal of my research (see https://arxiv.org/pdf/2210.07956.pdf, Equation 16).

Unfortunately, the whole results/proofs are split over 3 distinct papers and one book in Italian (for the "sfasamento" analysis), which extends the results we are currently talking about (see https://nntdm.net/volume-26-2020/number-3/245-260/, Conjecture 2 and Remark 5 derived from "La strana coda della serie n^n^...", 2011).

[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!".

[JmsNxn]

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 \(f_a\); is of the same type as this family.

[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.

[JmsNxn]

Yes. I remember you focused on \(10\)--which is useless in p-adic theory (it's called p-adic because p needs to be a prime). Where then you described \(2\) and \(5\) as the "atoms" of your investigation. But what you did for \(2\) and \(5\) can definitely be done for every \(p\).

Unfortunately, I am lacklustre at this. I can read and understand. But I cannot prove and be on the cutting edge. p-adic shit always confuses me, and I can't be on the forefront of it. But I hope you understand that you have carved out a fairly straight forward result. Which I'm happy to call Marco's result, or however you want to say it.

Marco's Theorem:

For all prime numbers \(p\), and all natural numbers \(a \in \mathbb{N}\); if we call \(^N a = a^{a^{...^a}}\) \(N\) times. Then the value:

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

Where \(\mathbb{Q}_p\) is the p-adic rational numbers.

---

From your papers this much is obvious, which is why I was surprised by your work. I would've never guessed this if you gave me a 1000 life times. Plus, I'm not that good at this shit, but I do know the general algebraic tools involved. I do believe this is your result. If anything, I am just changing some words around  

Regards, James

[marcokrt]

Yes. I remember you focused on \(10\)--which is useless in p-adic theory (it's called p-adic because p needs to be a prime). Where then you described \(2\) and \(5\) as the "atoms" of your investigation. But what you did for \(2\) and \(5\) can definitely be done for every \(p\).

Unfortunately, I am lacklustre at this. I can read and understand. But I cannot prove and be on the cutting edge. p-adic shit always confuses me, and I can't be on the forefront of it. But I hope you understand that you have carved out a fairly straight forward result. Which I'm happy to call Marco's result, or however you want to say it.

Marco's Theorem:

For all prime numbers \(p\), and all natural numbers \(a \in \mathbb{N}\); if we call \(^N a = a^{a^{...^a}}\) \(N\) times. Then the value:

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

Where \(\mathbb{Q}_p\) is the p-adic rational numbers.

---

From your papers this much is obvious, which is why I was surprised by your work. I would've never guessed this if you gave me a 1000 life times. Plus, I'm not that good at this shit, but I do know the general algebraic tools involved. I do believe this is your result. If anything, I am just changing some words around  

Regards, James

Yeah, I see... you are right (and too kind!), of course (about the name, it doesn't matter... here we have just an agreed starting point for a new research that belongs to everybody who is interested in the topic). My original thought is that we could repeat the very same thing done in those papers by choosing different numeral systems (not only radix-\(p\), where \(p\) is a prime), such as radix-\(6\). I believe that we can get another "Equation 16", with fewer lines and based only on the \(2\)-adic and \(3\)-adic valuation of a very simple "manipulation" of the base \(a\), and so forth. This would mean that we could abstractly solve the numeral system issue for an arbitrarily large number of cases... IMHO, a more powerful tool can achieve the final goal by induction, maybe (just to say).

About the "pure" \(p\)-adic approach, it would be the key to raise this research to the next level... it is a goal that I never set out to achieve, and it is a great intuition that you have shared here with us, so I really hope you will put it in a preprint or so, since it might actually be worth it in the future.
The result will be a totally different approach that I haven't set up in the trilogy and it will lead to new, exciting, results. Let's say, just a collection of what you have written here in a 4 page long preprint on ResearchGate and/or arXiv, would (IMHO) be a good starting point... you know more than me about how to go forward through \(p\)-adics, since \(\mathbb{Q} \subset \mathbb{Q}_p\) and, for any given degree, there are only finitely many field extensions of the aforementioned \(p\)-adic field at the end .

[JmsNxn]

Yes. I remember you focused on \(10\)--which is useless in p-adic theory (it's called p-adic because p needs to be a prime). Where then you described \(2\) and \(5\) as the "atoms" of your investigation. But what you did for \(2\) and \(5\) can definitely be done for every \(p\).

Unfortunately, I am lacklustre at this. I can read and understand. But I cannot prove and be on the cutting edge. p-adic shit always confuses me, and I can't be on the forefront of it. But I hope you understand that you have carved out a fairly straight forward result. Which I'm happy to call Marco's result, or however you want to say it.

Marco's Theorem:

For all prime numbers \(p\), and all natural numbers \(a \in \mathbb{N}\); if we call \(^N a = a^{a^{...^a}}\) \(N\) times. Then the value:

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

Where \(\mathbb{Q}_p\) is the p-adic rational numbers.

---

From your papers this much is obvious, which is why I was surprised by your work. I would've never guessed this if you gave me a 1000 life times. Plus, I'm not that good at this shit, but I do know the general algebraic tools involved. I do believe this is your result. If anything, I am just changing some words around  

Regards, James

Yeah, I see... you are right (and too kind!), of course (about the name, it doesn't matter... here we have just an agreed starting point for a new research that belongs to everybody who is interested in the topic). My original thought is that we could repeat the very same thing done in those papers by choosing different numeral systems (not only radix-\(p\), where \(p\) is a prime), such as radix-\(6\). I believe that we can get another "Equation 16", with fewer lines and based only on the \(2\)-adic and \(3\)-adic valuation of a very simple "manipulation" of the base \(a\), and so forth. This would mean that we could abstractly solve the numeral system issue for an arbitrarily large number of cases... IMHO, a more powerful tool can achieve the final goal by induction, maybe (just to say).

About the "pure" \(p\)-adic approach, it would be the key to raise this research to the next level... it is a goal that I never set out to achieve, and it is a great intuition that you have shared here with us, so I really hope you will put it in a preprint or so, since it might actually be worth it in the future.
The result will be a totally different approach that I haven't set up in the trilogy and it will lead to new, exciting, results. Let's say, just a collection of what you have written here in a 4 page long preprint on ResearchGate and/or arXiv, would (IMHO) be a good starting point... you know more than me about how to go forward through \(p\)-adics, since \(\mathbb{Q} \subset \mathbb{Q}_p\) and, for any given degree, there are only finitely many field extensions of the aforementioned \(p\)-adic field at the end .

Lmao! The most I could do is present your research to other mathematicians. And if you give me the clear to do such, I will. I cannot prove what I wrote; I can only say: This is probably what happens after looking at Marco's paper...

Please, understand, if you are giving me free reign on your result. That means I will talk about your result freely. And talk to other mathematicians freely. I will always give you credit for the atoms, the molecular understanding. But, I may consult far more advanced mathematicians. And in no way, am I trying to undermine your work. You have quite literally, done a \(\sqrt{2}\) is irrational kinda Pythagorean result. And I will always credit you   I ain't get where I am by stealing from people.  

But I do believe you have proved a deeper result than you realize. I don't know the result. But I know it's deeper than you've presented.....

If I present this to stronger and smarter mathematicians, I will always give you credit. And if anything, I'll give you more credit, than less credit. The way you are presenting yourself now, seems to hint at that. So, if I talk to mathematicians about p-adic shit, I'm just going to credit this shit to marco. Even though, I may have had a hand in the development of the language.

But, to be fair, it's your work

Even if you didn't prove Marco's Theorem, and technically I proved it, or someone else did. It's still Marco's Theorem. I just adapted the language and that's about it

---

Marco's work on \(p=2\) and \(p=5\) describe the affair for all \(p\). He just, additionally, added some finesse for \(2 \times 5 = 10\)....


Just know, when I present this to Higher level mathematicians; you get 90% of the credit

[marcokrt]

James, you are great and a very nice person... many thanks! I spent a dozen years struggling with the congruence speed formula, without (n)ever looking around for something bigger, as you did. I simply borrowed for a while the 10-adic argument in order to find Equation 16, while this would be a mere starting point.
Thus, you are free to share that papers and their "atoms" to any expert you wish (naming the \(p\)-adic theorems as you prefer, of course) and I am very excited very happy with this
Just in case, it would be an honour for me to be credited in some very advanced piece of art concerning those "atoms", but please also remember to give to yourself the proper credit!