Yesterday, 04:25 PM
(This post was last modified: Yesterday, 07:58 PM by marcokrt.
Edit Reason: Fixing LaTeX
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@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!
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!
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).