12/12/2011, 08:02 PM
(12/12/2011, 05:20 AM)JmsNxn Wrote: Sadly, I don't speak Italian, but looking at the math that seems very wild and interesting. The fact that the same digits reoccur in \( ^k 3 \) is very... weird?
I wonder, does this happen exclusively for base ten? Would this happen in binary or hexadecimal? I think it would... right?
That's very weird. This has to tell us something. Too bad I don't speak Italian to see what you've concluded.
Nonetheless; this takes the cake for "mathematical beauty of the day", maybe week, for me.
very cool find! I don't even know how you evaluate such large values of integer tetration of three.
Thank you. Yes... it's a general outcome.
BTW this represents only the beginning, I've explained every single rule concerning the convergence speed of a generic base. I've also found a lot of laws about the digits at the left of the convergent ones...
To calculate this digits, it's sufficient to use Wolfram|Alpha or others free programs
M
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)
