Number of stable digits of any integer tetration

TL;DR

This paper presents an exact formula to determine the number of stable, unchanging last digits in integer tetrations, revealing deep number-theoretic properties related to divisibility and valuations.

Contribution

It introduces a precise formula for computing stable digits in tetrations, linking it to p-adic valuations and congruence classes, extending previous understanding.

Findings

Exact formula for stable digits in tetration bases not coprime to 10

Maximum gap between bounds is related to the congruence speed V(a)

V(a) linked to 2-adic or 5-adic valuations depending on a's class

Abstract

In the present paper we provide a formula that allows to compute the number of stable digits of any integer tetration base $a\in\mathbb{N}_0$. The number of stable digits, at the given height of the power tower, indicates how many of the last digits of the (generic) tetration are frozen. Our formula is exact for every tetration base which is not coprime to $10$, although a maximum gap equal to $V(a)+1$ digits (where $V(a)$ denotes the constant congruence speed of $a$) can occur, in the worst-case scenario, between the upper and lower bound. In addition, for every $a>1$ which is not a multiple of $10$, we show that $V(a)$ corresponds to the $2$-adic or $5$-adic valuation of $a-1$ or $a+1$, or even to the $5$-adic order of $a^{2}+1$, depending on the congruence class of $a$ modulo $20$.