On the relation between perfect powers and tetration frozen digits

TL;DR

This paper explores the connection between perfect powers and tetration, revealing how certain perfect powers relate to tetration bases through their congruence properties and digit patterns.

Contribution

It introduces specific sets of perfect powers characterized by their congruence speed and establishes a link to tetration digit patterns, including the existence of infinitely many perfect powers with a given congruence speed.

Findings

Identifies a relation between perfect powers and tetration frozen digits.

Proves the existence of infinitely many perfect powers with a specified congruence speed.

Establishes a link between the degree of perfect powers and digit patterns in tetration.

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 $c$.