Gaussian Happy Numbers

TL;DR

This paper generalizes the concept of happy numbers to Gaussian integers, analyzing fixed points, cycles, and the existence of long arithmetic sequences, providing methods and results for various values of B.

Contribution

It extends the theory of happy numbers to Gaussian integers, characterizes fixed points and cycles, and explores the existence of long arithmetic sequences in this new setting.

Findings

Identified fixed points and cycles for small B values.

Provided a method to compute Gaussian B-happy numbers for any B ≥ 2.

Proved conditions for the existence of arbitrarily long arithmetic sequences.

Abstract

This paper extends the concept of a $B$-happy number, for $B \geq 2$, from the 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 $B \geq 2$. We discuss heights of Gaussian $B$-happy numbers, proving results concerning the smallest Gaussian $B$-happy numbers of certain heights. Finally, we prove conditions for the existence and non-existence of arbitrarily long arithmetic sequences of Gaussian $B$-happy numbers.