Tetration and p-adic

[marcokrt]

But, I fully believe that for natural numbers, Marco has shown that \(^\infty a\) for \(a \in \mathbb{N}\) is always in \(\mathbb{Q}_p\)--for every prime. And it is never irrational.

I think irrationality will happen for \(^N a\) and \(\mathbb{N}\); and \(N\)'s relation to the prime \(p\). But, these are still in \(\mathbb{Q}_p\). And I believe that Marco's result; if not proves this result, casts a wide net of results in which \(^\infty a \in \mathbb{Q}_p\). Rather than being irrational, in \(^\infty a \in \mathbb{R}_p / \mathbb{Q}_p\)...... The digits where we take Rational numbers, and tend to infinity, we see a repeatable pattern. Which looks like Marco's modular stuff. And studying that; talks about lists of digits, their repeating patterns, under tetration, under the norm where growth shrinks, and shrinking grows....

Wow, it does feel like becoming an eyewitness to a new frontier. 

My thought, exactly! There are so many things that you can do and my trilogy of papers is just a starting point looking to a new world made of discoveries and fascinating relations, not only involving the number of "convergent" digits peculiar of the integer tetration, but also the figures to their left... just like a wave in the ocean that propagates itself in the shape of concentric circles, gradually becoming more and more indistinguishable from the rest of the water (and finally leading us to some kind of caos theory approach, for the most significant digits, maybe?).
Now, just think to extend the aforementioned big world to a new dimension, by considering the reals or maybe taking into account the complex plane for a generic tetration base... we can get some surprising answers over there and I can only wish "Good luck to you all!".