Gaussian happy numbers
Breeanne Baker Swart, Susan Crook, Helen G. Grundman, Laura L. Hall-Seelig
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Source: Crossref
Published: Apr 1, 2022
DOI: 10.1216/rmj.2022.52.415
Open original source ↗Source 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.
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