Major references

[marcokrt]

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).

Hi Marco, forgive me for the typo. I'm actually well aware of your surname, I'm also italian so I don't have the excuse for being not used to accents. The reason for the error is, I was copy-pasting the name of the files in my folder on the forum, and windows do not accept special symbols naming the files then I had no enough time to double check the post.

Thanks for the contribution btw. I'll update my folder with your work. Some work on number theory related to hyperoperations is always welcome imho... since it is an underdeveloped topic. We need to face the problem from multiple viewpoints.

Cordialmente

ps: mi sono preso la liberta di usare i miei poteri da mod per editare il tuo ultimo messaggio e sistemare la sintassi LaTex, adesso aprendo il post puoi controllare come va fatta. You have to write "\ (" before a formula and "\ )" as closure.

I really do appreciate everything you have done, MphLee. Grazie!

[marcokrt]

@MphLee, just a brief bibliographic update to my December 2022 post.

On October 15, 2026, Gabriele Di Pietro and I published a new peer-reviewed paper in NNTDM: A Compact Notation for Peculiar Properties Characterizing Integer Tetration (Vol. 32, No. 3, pp. 477–496), https://doi.org/10.7546/nntdm.2026.32.3.477-496.
One of the main results, presented in Appendix B, is a general characterization of squarefree numeral systems: these are precisely the radices for which constant congruence speed is a universal structural property of all integer tetration bases greater than \(1\) that are not divisible by the radical of the radix.
The paper also describes a general method for deriving explicit congruence-speed formulas for any given squarefree radix \(r\), including composite ones, by studying the corresponding \(r\)-adic stationary solutions (and, as a worked example, we derive the complete formula for radix \(6\), following the same approach previously developed for radix \(10\)).
For the decimal case, the paper provides a framework for determining the full congruence-speed profile, including transient behavior before stabilization. We also prove a multiplicative lower bound: the constant congruence speed of a product is at least the minimum of those of its factors, provided the product is not divisible by \(10\) (see Theorem 2.1 and Appendix A).
Finally, we conjecture that the analogous inequality holds for all composite squarefree radices (Conjecture 3.1).
I thought it might be a useful addition to the bibliography discussed in this thread.

P.S.
Could you also correct the spelling of my surname in the bibliography, wherever necessary? It's "Ripà", not "Ripa".
Thanks in advance!