# The Powers of Two: An Introduction

An introduction and index to an article series on the properties of and patterns in the powers of two

The powers of two: 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024… Boredom by numbers, as I like to call them. The powers of two are often the only companions of my idle mind. They kept me company through slow internet connections, bumper-to-bumper traffic, long waiting lines and practically any of the most uneventful situations life can procure. I would have mental contests with myself to see how far into the sequence I could reach, and up to how many digits I could hold (or until the line finally moves).

But perhaps out of pity for me, more than being a simple mental exercise, these numbers have also shared some of their most exquisite and interesting patterns — from repeating digits to unique properties. While at a restaurant waiting for our orders to arrive, I fondly listed the powers of two on a stained tissue paper. When written down instead of painstakingly memorized, I noticed how the ones digit repeated periodically: 1, 2, 4, 8, 6, 1, 2, 4, 8, 6 and so on. Intrigued, I observed if a similar pattern would emerge for the ones and tens digits, and I was not disappointed, with the values repeating every 20 numbers. But with our orders served and my tissue paper more black with ink than its original white, I put my astonishment aside for now.

Later on, I did a bit of research on my personal ‘discovery,’ and found that this pattern not only repeats for the trailing n digits with a predictable period, but that other powers exhibit similar properties. Perhaps there was order in the chaos of exponential sequences. And perhaps there is intrigue to be found in the most dull of circumstances.

As both my research and curiosity grew, I decided to compile these small secrets the powers of two have imparted and shared through these articles — hopefully with the amazement I have felt and continue to feel along with it. Maybe someone would find the same or even more interest and astonishment from what the powers of two have to say, or perhaps the thoughts may simply brighten an otherwise boring day.

Or not. Well, to each his own.

All the python code that will be used throughout the article series can be found here for anyone’s perusal: https://github.com/JoaquindeCastro/The-powers-of-2