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!
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!