Tetration and p-adic

[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