Tetration and p-adic

[Ember Edison]

I'm suddenly curious if anyone in history has studied Tetration on  \(   \mathbb{Z}_p, \mathbb{Q}_p, \mathbb{C}_p       \).

And super-root, super-logarithm on  \(   \mathbb{C}_p      \).

Intuitively they should exist, but I never seem to have read the literature on them (and this forum).

[JmsNxn]

The only thing I've ever read towards P-adic numbers and tetration was posted here. I'm sure you can find it; the author of the post managed to publish his paper in a journal. But it was a more barebones version of P-adic work. I made some comments that this is definitely related to p-adic numbers. The post was about digit patterns in natural number tetrations. So if \(N\) is a natural number; and \(K\) is too; then \(^K N\) has common features in its digit patterns for base \(d\). Where he had some very interesting comments about choosing prime numbers (Other than 10). It was fairly recent so you should be able to find it. The common features to the digit patterns, is just the inner workings of P-adic analysis.

[Ember Edison]

The only thing I've ever read towards P-adic numbers and tetration was posted here. I'm sure you can find it; the author of the post managed to publish his paper in a journal. But it was a more barebones version of P-adic work. I made some comments that this is definitely related to p-adic numbers. The post was about digit patterns in natural number tetrations. So if \(N\) is a natural number; and \(K\) is too; then \(^K N\) has common features in its digit patterns for base \(d\). Where he had some very interesting comments about choosing prime numbers (Other than 10). It was fairly recent so you should be able to find it. The common features to the digit patterns, is just the inner workings of P-adic analysis.

yes, I got it. 

https://math.eretrandre.org/tetrationfor...p?tid=1374

[tommy1729]

There have been quite alot investigations but not many results.

In most cases it is just mod arithmetic and digit patterns.

Others have made conjectures that quickly got debunked.

Good nontrivial questions and good nontrivial answers are rare.

Sometimes things are not even well-defined.

I think all results are in this group , mathoverflow or mathstack, arxiv and vixra.

Most considerations were done by amateurs.

I second James comment, this is probably one of the best and almost only one we have here.

Most ideas got in the garbage can before even reaching the internet, including my own.

I am thinking about working on it though.

To avoid fragmentation and chaos, I will post it here when I do have a good idea, so you can find it easily.

regards

tommy1729

[marcokrt]

The only thing I've ever read towards P-adic numbers and tetration was posted here. I'm sure you can find it; the author of the post managed to publish his paper in a journal. But it was a more barebones version of P-adic work. I made some comments that this is definitely related to p-adic numbers. The post was about digit patterns in natural number tetrations. So if \(N\) is a natural number; and \(K\) is too; then \(^K N\) has common features in its digit patterns for base \(d\). Where he had some very interesting comments about choosing prime numbers (Other than 10). It was fairly recent so you should be able to find it. The common features to the digit patterns, is just the inner workings of P-adic analysis.

yes, I got it. 

https://math.eretrandre.org/tetrationfor...p?tid=1374

Dear Ember Edison, that reference is an old one, you can find the most recent results here: https://arxiv.org/abs/2208.02622 (see Section 2 for the g-adic/p-adic stuff) and https://arxiv.org/abs/2210.07956

I would be glad if somebody here could manage to simplify the given results in terms of p-adic (for any given squarefree numeral system, maybe).

[marcokrt]

P.S. The above are basically the same papers in LaTeX, so you can download the source TeX codes in order to write your own manuscript easily

[JmsNxn]

I would also love to see this in a p-adic sense! From my perspective it just smells like p-adic analysis so much. But I don't know enough Perhaps something on the lines of \(^\infty 2\) is a repeating fraction in \(\mathbb{Q}_2\) or something.

[Ember Edison]

Dear Ember Edison, that reference is an old one, you can find the most recent results here: https://arxiv.org/abs/2208.02622 (see Section 2 for the g-adic/p-adic stuff) and https://arxiv.org/abs/2210.07956

I would be glad if somebody here could manage to simplify the given results in terms of p-adic (for any given squarefree numeral system, maybe).

Thanks a lot, I'll try to understand it.

[JmsNxn]

Dear Ember Edison, that reference is an old one, you can find the most recent results here: https://arxiv.org/abs/2208.02622 (see Section 2 for the g-adic/p-adic stuff) and https://arxiv.org/abs/2210.07956

I would be glad if somebody here could manage to simplify the given results in terms of p-adic (for any given squarefree numeral system, maybe).

Thanks a lot, I'll try to understand it.

Hey, Ember. Remember that this paper is discovering a recursive pattern in the distribution of the digits \(^Na\); where upon this recursive pattern as \(N\to \infty\) repeats itself. And therefore we have something kind of p-adic. We have something like:

\[
^\infty a = \overline{(a_1a_2a_3...a_p)} A\\
\]

Where \(A\) is a fixed natural sequence of digits; as is \((a_1a_2a_3...a_p)\). But the latter continues on infinitely towards the left. Where, Marco's paper is shredding how to find \(a_j\) for all \(1 \le j \le p\).

Which just fucking smells like \(p\)-adic!  

Great job with your work, Marco! I've re read a bunch of your stuff. You're definitely on to something! I think there might be some clever trick that your missing. But still. Dope as fuck!

Regards, James

[Ember Edison]

I would also love to see this in a p-adic sense! From my perspective it just smells like p-adic analysis so much. But I don't know enough Perhaps something on the lines of \(^\infty 2\) is a repeating fraction in \(\mathbb{Q}_2\) or something.

I think we still need to understand the operation in question in terms of the power series. 

Using the exponential \(\exp_p(z)=\sum\limits_{n=0}^\infty\frac{z^n}{n!}\) and logarithmic functions \(\log_p(1+x)=\sum\limits_{n=1}^\infty \frac{(-1)^{n+1}x^n}{n}\) defined on \(\mathbb{C}_p\), We can obtain binary functions \(a^b\) and \(log_a(b)\) that are valid for any p-adic. 

but I have absolutely no idea about binary functions \(sexp(b,h),\ slog(b,z),\ ssqrt(h,z)\) that are valid for any p-adic!