Graham's number stable digits: An exact solution

TL;DR

This paper precisely determines the stable digits of Graham's number and related tetrations in base 3, revealing exact digit patterns and differences for large hyperexponents.

Contribution

It provides an exact characterization of the stable digits of Graham's number and related tetrations, extending understanding of their digit structure in base 3.

Findings

The last stable digits of Graham's number are determined by its super-logarithm.

The $ ext{slog}_3(G)$-th rightmost digit of Graham's number differs from that of higher tetrations.

The $ ext{slog}_3(^{n}3)$-th least significant digit of the difference is 4.

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 $\operatorname{slog}_3(G) - 1$ rightmost digits (where $\operatorname{slog}$ indicates the integer super-logarithm). This is an exact result since the $\operatorname{slog}_3(G)$-th rightmost digit of $G$ differs from the $\operatorname{slog}_3(G)$-th rightmost digit of $^{n+1}3$. Furthermore, we show that the $\operatorname{slog}_3(^{n}3)$-th least significant digit of the difference between Graham's number and any base $3$ tetration whose integer hyperexponent exceeds $n$ is $4$.