On the relation between perfect powers and tetration frozen digits
Marco Ripà
TL;DR
This work addresses the problem of connecting the digit-pattern patterns in rapidly growing power towers (tetration) with classical perfect powers. It introduces and formalizes the constant congruence speed $V(a)$ of tetration bases in radix-10, and proves, via constructive digit-based techniques and p-adic tools, that for every positive integer $c$ there exist infinitely many $c$-th perfect powers with constant congruence speed $c$. The core methods combine automorphic/10-adic considerations with the LTE lemma to compute valuations of expressions like $(10^{k+t}+10^t+1)^c-1$, enabling explicit base constructions such as $(10^{k+t}+10^{t-\min\{\nu_5(c),\nu_2(c)\}}+1)^c$ that realize the target speed. The results yield a rich family of infinite, explicitly described sets of perfect powers tied to tetration digit stability, with potential implications for modular arithmetic of power towers and related sequences.
Abstract
This paper provides a link between integer exponentiation and integer tetration since it is devoted to introducing some peculiar sets of perfect powers characterized by any given value of their constant congruence speed, revealing a fascinating relation between the degree of every perfect power belonging to any congruence class modulo $20$ and the number of digits frozen by these special tetration bases, in radix-$10$, for any unit increment of the hyperexponent. In particular, given any positive integer $c$, we constructively prove the existence of infinitely many $c$-th perfect powers that have a constant congruence speed of $c$.