Graham's number stable digits: an exact solution (2411.00015v1)

Abstract: In the decimal numeral system, we prove that the well-known Graham's number, $G := ! ^{n}3$ (i.e., $3^{3^{\cdot^{\cdot^{\cdot^{3}}}}} n$-times), and any base $3$ tetration whose hyperexponent is larger than $n$ share the same $slog_3(G) - 1$ rightmost digits (where $slog$ indicates the integer super-logarithm). This is an exact result since the $slog_3(G)$-th rightmost digit of $G$ is different from the $slog_3(G)$-th rightmost digit of $^{n+1}3$. Furthermore, we show that the $slog_3(^{n}3)$-th least significant digit of the difference between Graham's number and any base $3$ tetration whose integer hyperexponent is beyond $n$ is $4$.

• The paper determines the exact number of stable rightmost digits of Graham's number using base-3 tetration and the super-logarithm function.
• It asserts that Graham's number has slog_3(G) - 1 stable rightmost digits, coinciding with smaller base-3 power towers up to a specific threshold.
• The findings provide insights into digital stability in hyperlarge numbers and could influence research in number theory and computational mathematics.

Examination of Stable Digits in Graham's Number

The paper "Graham's Number Stable Digits: An Exact Solution" by Marco Rip investigates specific properties of Graham's number, with a focus on determining patterns in its stable digits within the decimal system. Graham's number, denoted as $G$, is famously large, introduced as an upper bound in a Ramsey Theory-related problem, and is known for its formidable size, making it infeasible to represent fully using conventional digital means.

This research presents a methodical analysis of Graham's number in terms of its digital stability using base-3 tetration and the super-logarithm function. The paper posits that the rightmost digits of Graham's number in the decimal system, denoted by $slog_3(G) - 1$, coincide with those of any base-3 power tower whose hyperexponent exceeds a specific threshold $n$. The paper further asserts that the $slog_3(G)$-th rightmost digit of Graham's number diverges from that of the next power tower iteration.

Main Contributions and Theoretical Insights

The key theoretical contribution lies in the assertion that $slog_3(G)$-th rightmost digits, under base-3 tetration, demonstrate a repeating stability up to a point and then experience a switch in congruence class. This proposition challenges prior assumptions regarding digital stability in hyperlarge numbers.

Several theorems and propositions are introduced:

1. The method to determine stable digits through the super-logarithm function provides a fresh perspective on quantifying digital stability in Graham's number.
2. Definitions and expansions on concepts such as constant congruence speed and phase shift illuminate their roles in characterizing the behavior of digital sequences in tetration results.
3. The asymptotic phase shift analysis, particularly concerning the tetration base-3, systematically categorizes phases shifts arising from alterations in the hyperexponent.

Numerical Results and Implications

The paper yields certain fixed numerical findings, specifically regarding the stability of digits in sequences generated through high tetrations involving the base number 3. The assertion that Graham's number has $slog_3(G) - 1$ stable digits could broadly facilitate understanding how such numbers, represented in multiple radix systems, have parts of their sequences unaffected by further growth in power expansions.

The work could influence research in number theory addressing large number properties and their applications across computational mathematics. While directly grounded in theoretical inquiry, such knowledge has prospective implications for algorithms where number size and behavior predictability are crucial, for instance, in cryptographic applications or complexity theory.

Future Research Directions

The approach in mapping digital sequences and stability in exceedingly large numbers could be expanded to other bases and mathematical constructs beyond Graham's number. Additionally, exploring the implications of stable digit sequences might enrich digital signal processing, where vast data sets require invariant segment identification for accuracy and efficiency.

In conclusion, the paper not only deepens our understanding of Graham's number and its unique position within Ramsey Theory but also fosters significant discourse on how extremely large numerical structures can be dissected to reveal inherent order, refining both theoretical knowledge and practical capabilities with hyperlarge numbers.