02/14/2023, 04:33 AM
(This post was last modified: 02/14/2023, 04:36 AM by marcokrt.
Edit Reason: minor edits
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(02/12/2023, 10:49 AM)Ember Edison Wrote:(02/12/2023, 03:45 AM)JmsNxn Wrote: 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).
Let \(G(n)\) be a generic reverse-concatenated sequence. If \(G(1) \notin \{2, 3, 7\}\), then \(^{G(n)}G(n) \pmod {10^d}≡^{G({n+1})}G({n+1}) \pmod {10^d}\), \(\forall n \in \mathbb{N}-\{0\}\)
("La strana coda della serie n^n^...^n", p. 60).
("La strana coda della serie n^n^...^n", p. 60).