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!
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)
