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!
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 \(G(1) \notin \{2, 3, 7\}\), 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).
