(12/20/2011, 11:29 PM)nuninho1980 Wrote: I understand that.
2^^3 = 16
2^^4 = 65536
2^^5 = 20035... 19718 digits ...156736
2^^6 = ???... ?.??x10^19727 digits ...8736
2^^7 = ???... ?.??x10^(?.??x10^19727) digits ...48736
4^^2 = 256
4^^3 = 13407... 145 digits ...84096
4^^4 = ???... ?.??x10^153 digits ...896
4^^5 = ???... 10^^2^153 digits ...8896
4^^6 = ???... 10^^3^153 digits ...28896
Every base is characterized by this kind of convergence... (sometimes) there are only a few steps without convergence.
The asymptotic convergence speed is constant for every base (the proof is in my book)!
If you like a little more fun, you can take a look at this (in the book I have called it "sfasamento"):
[5^10^i](mod 10^30):
0- 5
1- 9765625
2- 064351090230047702789306640625
3- 927874558605253696441650390625
4- 768305384553968906402587890625
5- 423444294370710849761962890625
6- 649817370809614658355712890625
7- 838703774847090244293212890625
8- 125944280065596103668212890625
9- 648495816625654697418212890625
10- 388659619726240634918212890625
11- 255141400732100009918212890625
12- 404334210790693759918212890625
13- 333762311376631259918212890625
14- 378043317236006259918212890625
15- 820853375829756259918212890625
16- 248953961767256259918212890625
… …
0- 5
1- 9765625
2- 064351090230047702789306640625
3- 927874558605253696441650390625
4- 768305384553968906402587890625
5- 423444294370710849761962890625
6- 649817370809614658355712890625
7- 838703774847090244293212890625
8- 125944280065596103668212890625
9- 648495816625654697418212890625
10- 388659619726240634918212890625
11- 255141400732100009918212890625
12- 404334210790693759918212890625
13- 333762311376631259918212890625
14- 378043317236006259918212890625
15- 820853375829756259918212890625
16- 248953961767256259918212890625
… …
... I've discovered different kinds of convergence/pseudo-convergence... it's related to caos theory too (the underlying mathematics is group theory by Galois).
Marco
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)
