Congruence speed of tetration bases ending with $0$

Abstract

For every non-negative integer $a$ and positive integer $b$, the congruence speed of the tetration $^{b}a$ is the difference between the number of the rightmost digits of $^{b}a$ that are the same as those of $^{b+1}a$ and the number of the rightmost digits of $^{b-1}a$ that are the same as those of $^{b}a$. In the decimal numeral system, if the given base $a$ is not a multiple of $10$, as $b:=b(a)$ becomes sufficiently large, we know that the value of the congruence speed does not depend on $b$ anymore, otherwise the number of the new rightmost zeros of $^{b}a$ drastically increases for any unit increment of $b$ and, for this reason, we have not previously described the congruence speed of $a$ when it is a multiple of $10$. This short note fills the gap by giving the formula for the congruence speed of the mentioned values of $a$ at any given height of the hyperexponent.