Frozen digits in any integer tetration marcokrt Junior Fellow Posts: 12 Threads: 3 Joined: Dec 2011 08/13/2022, 07:18 PM Happy to announce that me and Luca (Luknik) have finally published the paper with the direct map of the constant congruence speed of tetration: Number of stable digits of any integer tetration Basically, assuming radix-$10$, for any base $a \in \mathbb{N}_{0}$ which is not a multiple of $10$ and considering a unitary increment of a sufficiently large hyperexponent $b \in \mathbb{Z}^{+}$, we can find the number of new stable (i.e., previously unfrozen) digits of ${^{b}a$ by simply taking into account the $2$-adic or the $5$-adic valuation of $a \pm 1$, or the $5$-adic valuation of $a^2+1$ (see Equation 16). The above is my third and last paper on this fascinating and peculiar property of tetration. Everything was inspired by the intriguing open field that I started to discover thanks to the registration of this forum almost $11$ years ago. Thank you everybody, feedback is welcome! Let G(n) be a generic reverse-concatenated sequence. If G(1)≠{2, 3, 7}, [G(n)^^G(n)](mod 10^d)≡[G(n+1)^^G(n+1)](mod 10^d), ∀n∈N\{0} (La strana coda della serie n^n^...^n, 60). bo198214 Administrator Posts: 1,615 Threads: 101 Joined: Aug 2007 08/13/2022, 08:27 PM (This post was last modified: 08/13/2022, 08:28 PM by bo198214.) Yippee, Champaign! ? JmsNxn Ultimate Fellow Posts: 993 Threads: 117 Joined: Dec 2010 08/14/2022, 04:51 AM (This post was last modified: 08/14/2022, 04:53 AM by JmsNxn.) Absolutely Beautiful, Marco! Super proud and happy for you! I'm excited to read this. It's not a problem I'm particularly interested in, but this is awesome! And yes, like Bo said, have some Champagne on me! « Next Oldest | Next Newest »

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