Happy Numbers: An Exploration of An Iterated Function in Different Bases
Samantha Alison Mutter · 2010
This project is concerned with happy and sad numbers. To decide whether a number is happy or sad, you break it up into its individual digits, square each one, and add them all together. Then you take this result, and repeat the process. If you repeat the process enough times, you will eventually reach either 1, in which case your initial number is happy, or a set of 8 numbers that cycle endlessly, in which case it is sad. This fact is only true when we are working with numbers from the base-10 system that we use everyday. In other bases, such as the binary system that is used in computing, the function displays other cyclical behavior, such as cycles of varying lengths and fixed points other than 1. For some set bases and initial points, we can predict the eventual behavior of the function over time.