Regularity versus complexity in the binary representation of 3^n

TL;DR

This paper investigates the binary representation of powers of 3, revealing regular patterns and structures explained by 2-adic numbers, and extends these observations to powers of other bases.

Contribution

It introduces a novel perspective linking binary patterns of powers to 2-adic number theory, uncovering regularities in seemingly disordered bit patterns.

Findings

Diagonal stripes in bits of 3^(2^n) identified

Regularity explained via 2-adic power series

Patterns extend to base-p representations of k^(p^n)

Abstract

We use the grid consisting of bits of 3^n to motivate the definition of 2-adic numbers. Specifically, we exhibit diagonal stripes in the bits of 3^(2^n), which turn out to be the first in an infinite sequence of such structures. Our observations are explained by a 2-adic power series, providing some regularity among the disorder in the bits of powers of 3. Generally, the base-p representation of k^(p^n) has these features.