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Regularity versus Complexity in the Binary Representation of 3*n*

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

#### 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^{2n}, 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*^{pn} has these features.