Gaussian happy numbers

Abstract

This paper extends the concept of a $B$-happy number, for $B\ge 2$, from the positive rational integers, ${\mathbb{Z}}^{+}$, to the Gaussian integers, $\mathbb{Z}[i]$. We investigate the fixed points and cycles of the Gaussian $B$-happy functions, determining them for small values of $B$ and providing a method for computing them for any $B\ge 2$. We discuss heights of Gaussian $B$-happy numbers, proving results concerning the smallest Gaussian $B$-happy numbers of certain heights, and we prove conditions for the existence and nonexistence of arbitrarily long arithmetic sequences of Gaussian $B$-happy numbers. Finally, we consider an alternative definition of Gaussian happy numbers using expansions in base $-1+i$.