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arXiv · 2408.12257

Fixed Points and Cycles of the Kaprekar Transformation: 2. Even bases

Anthony Kay·Katrina Downes-Ward
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Abstract

We develop a classification of the fixed points and cycles of the Kaprekar transformation in even bases. The most numerous fixed points and cycles are those we denote symmetric and almost-symmetric; the structure of the cycles of these classes in base $b$ is determined by subgroups and cosets in the multiplicative group modulo $b-1$. We provide methods and formulae for enumerating the fixed points and cycles of these and other classes. A detailed survey of the fixed points and cycles is provided for bases 4, 6 and 8, including a rigorous proof that the classification is complete in base 4.

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BibTeXRIS

Anthony Kay, Katrina Downes-Ward. 2024-08-22. Fixed Points and Cycles of the Kaprekar Transformation: 2. Even bases. https://arxiv.org/abs/2408.12257

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