03/14/2026, 01:43 PM
(03/23/2025, 10:52 PM)MphLee Wrote: I'd like move this thread to the main section. I don't clearly see how this can belong specifically to the " hyperoperations and Related Studies";
About your claim "tetration is the only hyperoperator that is characterized by a constant congruence speed", is anything known about the congruence speed or of the behaviour of the p-mod reduction of higher hyperoperations?
Just spotted this (sorry for the delayed reply).
In the decimal system (assume the radix \(r=10\)) I proved that the congruence speed of tetration is eventually constant for every integer base \(a\) greater than \(1\) and not divisible by \(10\).
For positive integer bases \(a\) divisible by \(10\), instead, I derived the explicit formula for the number of new frozen digits when the height \(b\) of the power tower increases by one unit; in this case the growth is exponential and therefore the congruence speed is not constant.
From this observation it is quite natural that the phenomenon cannot extend to higher hyperoperations. If the number of frozen digits were constant for some hyperoperation, then for the next hyperoperation the number of frozen digits would grow extremely fast with the height \(b\) and could not stabilize.
Moreover, we have recently proven that the same result holds (at least) in every squarefree numeral system of radix \(r\) for each integer tetration base \(a>1\) not divisible by the radical of \(r\).
Therefore, assuming \(r>1\) squarefree, tetration is the only hyperoperation exhibiting a constant congruence speed for all positive integers not divisible by \(r\).
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).
