Regularity versus Complexity in the Binary Representation of 3n

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 32 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 kp n has these features.