"Gaussian Happy Numbers"
Authors: Swart, Breeanne Baker; Crook, Susan; Grundman, Helen G.; Hall-Seelig, Laura L.
Source: Rocky Mountain Journal of Mathematics, Volume: 52, Issue: 2, Pages: 415-429, DOI: 10.1216/rmj.2022.52.415, April 2022
Type of Publication: Article
Abstract: This paper extends the concept of a B-happy number, for B≥2, from the positive rational integers, Z+, to the Gaussian integers, 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≥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.