Graham's number stable digits: An exact solution

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

9 pages, 1 figure