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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| Accès en ligne: | https://arxiv.org/abs/2408.12257 |
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| _version_ | 1866914920741732352 |
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| author | Kay, Anthony Downes-Ward, Katrina |
| author_facet | Kay, Anthony Downes-Ward, Katrina |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_12257 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Fixed Points and Cycles of the Kaprekar Transformation: 2. Even bases Kay, Anthony Downes-Ward, Katrina Combinatorics Number Theory 11A99 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. |
| title | Fixed Points and Cycles of the Kaprekar Transformation: 2. Even bases |
| topic | Combinatorics Number Theory 11A99 |
| url | https://arxiv.org/abs/2408.12257 |