Tetration and hyperoperators


Do you know what an hyperoperator is?
If yes, it would be very easy to solve the analogy below.

Zeration:0=Addition:1=Multiplication:2=Exponentiation:3=???:4=…

I think the easiest way to undersand a little more about this strange world is to read a couple of Wiki pages:
Tetration

Now, do you wish to know more about this topic? Exploring deeper and deeper the p-adic convergence of the less significative digits of big numbers combined through tetration? Calculating by hands n^^i (mod 10), (mod 100), mod(1000)?

If you know a little bit of Italian, this could be the right book for you:
La strana coda della serie n^n^…^n

Otherwise, a great website to learn number theory applied to hyperoperators is this one:
Tetration.org

A few examples of the content are as follows (this is what, in the book, I have called “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 is related to caos theory too (the underlying mathematics is group theory by Galois).

There is much more, let’s say, we can construct some bases with an unlimited convergence speed, for example, 999…9. The number of “9″ (the lenght in digits of the base) gives us an equal “convergence speed in a single step”: i.e. [9999999^^n](mod 10^(7*n))==[9999999^^(n+1)](mod 10^(7*n)).

And, dulcis in fundo, a self-referential citation about Smarandache sequences and tetration [“La strana coda della serie n^n^…^n”, pag. 60]:

Let G(n) be a generic reverse-concatenated sequence. If G(1)≠2, [G(n)^^G(n)](mod 10^d)≡[G(n+1)^^G(n+1)](mod 10^d), ∀n∈N{0}.

A wonderful universe full of open problems and challenging adventures is waiting for brave amateurs. Good luck!

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