Major references

[marcokrt]

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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  

Thank you for your interest, I would be very glad if you (or anyone else on this forum) will publish further results about the congruence speed concept, generalizing what I have written and going beyond.
Basically, my starting idea is just to "count" how many digits frozes each unitary increment of the hyperexponent (and we could apply this also to hyper-\(3\) or hyper-\(5\), in general we have a function of the base and the (hyper)-exponent, which turns to not depend on the (hyper)-exponent under certain circumstances).
Then, by assuming radix-\(10\) and that the base is a positive integer which is not congruent to \(0\) modulo \(10\) by hypotesis, everything comes from the \(15\) solutions of the fifth degree equation \(y^t=y\) in the ring of decadic integers, since \(10\) is not a prime.
Now, since I am just a self-taught amateur in number theory, I am aware that the behaviour of \(p\)-adic numbers is "good" and smooth if compared to the behaviour of \(g\)-adic ones (where \(g\) is not prime). In general, I strongly believe that we can reply what we already did in "Number of stable digits of any integer tetration" (see equation 16) for any \(g\) that is a squarefree semiprime and also if \(g\) is a generic squarefree composite number, maybe... but it would be a great step forward if you can write something that shows this by using only \(p\)-adic analysis, best wishes!