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 \(c\).
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 \(c\).
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)
