(12/26/2022, 08:40 AM)marcokrt Wrote: .........
Awesome list! I would like to just point out that my real surname is "Ripà" (instead of "Ripa", a very common mistake indeed), mentioning also the last paper that I have written together with the other TetrationForum user "Luca Onnis", entitled "Number of stable digits of any integer tetration", which completes the trilogy on the congruence speed that I started by submitting to Notes on Number Theory and Discrete Mathematics, in 2019, the manuscript entitled "On the constant congruence speed of tetration" (see NNTDM, Vol. 26(3), pp. 245-260, DOI: 10.7546/nntdm.2020.26.3.245-260).
After that, I finally managed to provide an inverse map of this new function (assuming that radix-10 is given by hypothesis), the congruence speed of any integer tetration with the base which is not a multiple of 10, by publishing the mentioned paper entitled "The congruence speed formula" (available also on the arXiv at https://arxiv.org/abs/2208.02622).
Then, me and Luca have finally provided also the direct map of the "congruence speed of tetration" by considering the valuation function (applied to a few very simple manipulations of the given base) of the divisors of the squarefree value of the considered numerical system (i.e., 2 and 5, since 10 = 2 · 5) thanks to the paper titled "Number of stable digits of any integer tetration", which closes the bounds that I previously gave in "The congruence speed formula", by providing extended proofs based on the theorems published in "The congruence speed formula", so I think that it would be the best to mention both "The congruence speed formula" and "Number of stable digits of any integer tetration" (arXiv version: https://arxiv.org/abs/2210.07956) in order to provide the full map of this peculiar property of the integer tetration, named "constant congruence speed".
Just my two cents.
I am very interested in your work; as it deals with digit analysis. You've uncovered a general structure that the digts play. Have you ever tried \(\sqrt{2}\), and dealing with similar modular results? As an analyst myself; I tend to not be so worried about the digit patterns that appear in \(^52\). But if such a digit pattern were to appear in \(\sqrt{2}\), this would define an algebraic result.
Not to spoil what you are doing. I have followed your posts closely. I suggest reading about \(p\)-adic analysis. I cannot reduce your results to \(p\)-adic results. But for fucks sakes; it smells like it. There is a \(p\)-adic interpretation of your result. I do not know it; but I could probably work a guess. In this language you should find a clearer version of your formula. Not to degrade your result; you have done great work. Just to suggest--I believe we can transplant this idea.
Either way; I apologize if I'm being presumptuous. I'm just trying to help
