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
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).
