Ternary Digits of Powers of Two

TL;DR

This paper investigates the distribution of ternary digits in powers of two and the logarithm base 3 of 2, providing computational evidence supporting their uniform distribution and normality conjectures.

Contribution

It offers computational results up to 10^6 digits supporting the conjecture of uniform distribution and normality in these sequences.

Findings

Computational evidence supports uniform distribution of ternary digits in 2^n.

Strong evidence suggests log_3(2) is normal to base 3.

Implications discussed include connections to Benford's Law and Baker's Theorem.

Abstract

The \textit{ternary digits of $2^n$} are a finite sequence of 0s, 1s, and 2s. It is a natural question to ask whether the frequency of any string of 0s, 1s, and 2s in this sequence approaches the same limit for all strings of the same length, as the exponent $n$ approaches infinity (\textit{Uniform Distribution in the limit}). Currently the answer to this question is unknown. Even a much weaker conjecture by Erd\"os is still open. But we present computational results (up to $n = 10^6$) supporting uniform distribution in the limit. In this context, we discuss implications of Benford's Law and a special case of Baker's Theorem. Then we investigate the infinite sequence of ternary digits of $\log_3(2)$. There are analogous questions about the distribution of strings of 0s, 1s, and 2s in that sequence. If there is uniform distribution in the limit, then $\log_3(2)$ is called \textit{normal to base 3}. In the absence of definitive results, we can offer again computational evidence from the first $10^6$ ternary digits of $\log_3(2)$, strongly supporting the conjecture that $\log_3(2)$ is normal to base 3.