Regularity versus Complexity in the Binary Representation of 3n

Abstract

We use the grid consisting of bits of 3 n to motivate the defini- tion 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 reg- ularity among the disorder in the bits of powers of 3. Generally, the base-p representation of k p n has these features.