Number of stable digits of any integer tetration

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 $20$.

• The paper derives a precise formula for determining the number of stable digits in integer tetration.
• It employs p-adic valuations and congruence class analysis to establish rigorous upper and lower bounds for digit stability.
• The study introduces the concept of congruence speed to quantify the rate at which the digits stabilize for bases not coprime to 10.

Summary of "Number of stable digits of any integer tetration"

This paper, authored by Marco Ripa and Luca Onnis, presents an analytical framework for determining the number of stable digits in integer tetration, where the base $a$ is an integer not coprime to 10. Specifically, it tackles a niche problem in number theory involving the so-called "stable" last digits of tetrated numbers. These are the digits that remain consistent across further iterations of exponentiation. The paper introduces and elaborates on a formula tailored for this purpose, revealing intricate relationships between stable digits, congruence classes, and p-adic valuations.

Key Concepts and Results

1. Stable Digits and Tetration: The study focuses on identifying how many of these last digits, known as stable digits, remain unchanged at varying heights of the power tower. The emphasis is placed not merely on identifying these digits but on providing rigorous, exact upper and lower bounds for them.
2. Formula Derivation: The authors derive a mathematical formula that computes the stable digits of a given integer tetration for bases that are not multiples of 10. This formula accounts for constants like the 2-adic and 5-adic valuations dependent on the specific congruence class of $a$ modulo 20.
3. Congruence Speed: The notion of congruence speed ($V(a)$) is pivotal to their work. It serves as an indicator of the rate at which the digits stabilize. The paper breaks this down further by introducing the "constant congruence speed," particularly focusing on scenarios contingent on $a$'s properties with respect to divisibility and coprimality.
4. Adic Valuations: Detailed consideration of p-adic valuations underpins much of the theoretical development in handling numbers modulo powers of ten. They analyze how these valuations contribute to determining the number of stable digits, offering both insights and calculations for handling specific congruence classes.
5. Bounds and Exceptions: For base values that are multiples of particular small integers and coprimes to 10, exceptions are detailed with rigorous conditions. The authors draw from previously established literature to expand on cases where expected valuation calculations do not straightforwardly apply.

Theoretical and Practical Implications

Theoretically, this work advances our understanding of integer tetration in number theory, particularly within modular arithmetic and digit stability. It connects stable digit phenomena with deep properties of congruence and valuation, thus offering a structured approach to predicting tetration outcomes across numerical bases.

Practically, this exploration could impact computational mathematics, especially in areas requiring high precision such as cryptography and computational number theory. Understanding stable digit calculations can potentially streamline algorithms that handle large number computations by mitigating the need for exhaustive calculations through intelligent prediction of stable digits.

Speculation on Future Developments

Future research might expand this analytical framework to include bases that are multiples of 10 or incorporate more generalized congruence conditions. Further, interdisciplinary applications across cryptography and computational optimizations seem plausible avenues, potentially using these findings to enhance algorithms concerning large integer calculations.

This paper thus contributes a precise, technical exploration of a complex topic within number theory, enriching our understanding of tetration stability and its practical computations. As the work builds upon foundational number theory principles, it provides a stepping stone for both theoretical exploration and application-oriented innovation in connected fields.