02/20/2023, 12:07 PM
Thank you very much, dear James... yeah, I've definitely proven that stuff over the nonnegative integers, since it was the original goal of my research (see https://arxiv.org/pdf/2210.07956.pdf, Equation 16).
Unfortunately, the whole results/proofs are split over 3 distinct papers and one book in Italian (for the "sfasamento" analysis), which extends the results we are currently talking about (see https://nntdm.net/volume-26-2020/number-3/245-260/, Conjecture 2 and Remark 5 derived from "La strana coda della serie n^n^...", 2011).
Unfortunately, the whole results/proofs are split over 3 distinct papers and one book in Italian (for the "sfasamento" analysis), which extends the results we are currently talking about (see https://nntdm.net/volume-26-2020/number-3/245-260/, Conjecture 2 and Remark 5 derived from "La strana coda della serie n^n^...", 2011).
For every radix \(r\geq 2\), the congruence speed of tetration stabilizes at a positive constant for all bases \(a>1\) with \(\mathrm{rad}(r )\nmid a\) if and only if \(r\) is squarefree. (NNTDM 32(3), Appendix B)