Abstract
This paper extends the concept of a -happy number, for , from the positive rational integers, , to the Gaussian integers, . We investigate the fixed points and cycles of the Gaussian -happy functions, determining them for small values of and providing a method for computing them for any . We discuss heights of Gaussian -happy numbers, proving results concerning the smallest Gaussian -happy numbers of certain heights, and we prove conditions for the existence and nonexistence of arbitrarily long arithmetic sequences of Gaussian -happy numbers. Finally, we consider an alternative definition of Gaussian happy numbers using expansions in base .
Citation
Breeanne Baker Swart. Susan Crook. Helen G. Grundman. Laura L. Hall-Seelig. "Gaussian happy numbers." Rocky Mountain J. Math. 52 (2) 415 - 429, April 2022. https://doi.org/10.1216/rmj.2022.52.415
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