Complex Systems

Regularity versus Complexity in the Binary Representation of 3n Download PDF

Eric S. Rowland
Mathematics Department
Tulane University
New Orleans, LA 70118, USA

Abstract

We use the grid consisting of bits of 3n to motivate the definition of 2-adic numbers. Specifically, we exhibit diagonal stripes in the bits of 32n, 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 kpn has these features.