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12/11/2024, 08:48 PM
(This post was last modified: 12/24/2024, 05:28 PM by marcokrt.
Edit Reason: Adding one missed close parenthesis in the formula
)
In my recent paper entitled "On the relation between perfect powers and tetration frozen digits", I have established a new link between hyper-\(3\) (i.e., exponentiation) and hyper-\(4\) (i.e., tetration) by providing infinitely many tetration bases that are \(c\)-th perfect powers whose constant congruence speed is equal to \(c\). Now, the constant congruence speed of every integer greater than \(1\) and not a multiple of \(10\) is a positive integer that describes a peculiar feature of the given tetration base... and tetration is the only hyperoperator that is characterized by a constant congruence speed (see Definitions 1.1 and 1.2 of Number of stable digits of any integer tetration) for any nontrivial base (e.g., hyper-\(3\) has a constant congruence speed only for the multiples of \(10\), while pentation shows a constant "congruence acceleration" instead of constant "congruence speed").
My paper directly proves the existence of infinitely many positive integers that are perfect powers of degree \(c\) (exactly \(c\)) and whose constant congruence speed is equal to \(\min\{\nu_2({c}), \nu_5({c})\} + 2\), \(\min\{\nu_2({c}), \nu_5({c})\} + 3\), \(\min\{\nu_2({c}), \nu_5({c})\} + 4\), and so forth (where \(\nu_2({c})\) and \(\nu_5({c})\) indicate the \(2\)-adic order and the \(5\)-adic order of \(c\), respectively).
Denoting as \(V(a)\) the constant congruence speed of \(a\) (as usual), the fundamental equation to achieve this powerful result is
\(V((10^{k + t} + 10^{t - \min\{\nu_2({c}), \nu_5({c})\}} + 1)^c) = t\) (\(k,t \in \mathbb{N} : t > \min\{\nu_2({c}), \nu_5({c})\} + 1\)), which is true for any given positive integer \(c\).
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).
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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).
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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?
Mother Law \(\sigma^+\circ 0=\sigma \circ \sigma^+ \)
\({\rm Grp}_{\rm pt} ({\rm RK}J,G)\cong \mathbb N{\rm Set}_{\rm pt} (J, \Sigma^G)\)
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(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).
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Just a brief update: the paper mentioned in my previous post, " A Compact Notation for Peculiar Properties Characterizing Integer Tetration", has just been published in its final (peer-reviewed) version in Notes on Number Theory and Discrete Mathematics, 32(3) (2026), 477–496.
In particular, Appendix B establishes that squarefree radices are precisely those for which the congruence speed of integer tetration eventually stabilizes to a positive constant for every integer base greater than \(1\) not divisible by the radical of the radix.
The paper also investigates the behavior for non-squarefree radices and proves the following multiplicative lower-bound property, at least for the decimal numeral system:
\(V(x \cdot y) \geq \min\{V(x),V(y)\}\),
for all integers \(x,y>1\) such that \(10 \nmid x \cdot y \).
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).
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