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arXiv · 2402.07929

Congruence speed of tetration bases ending with $0$

Marco Ripà
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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.

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Marco Ripà. 2024-02-06. Congruence speed of tetration bases ending with $0$. https://arxiv.org/abs/2402.07929

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