Tetration and p-adic

[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.

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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 .