10/21/2026, 03:37 PM
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 \(10 \nmid x \cdot y \).
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 \(10 \nmid x \cdot y \).
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)
