On the relation between perfect powers and tetration frozen digits

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 $c$.