Midy's Theorem in non-integer bases and divisibility of Fibonacci numbers
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866918524161622016 |
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| author | Masáková, Zuzana Pelantová, Edita |
| author_facet | Masáková, Zuzana Pelantová, Edita |
| contents | Fractions $\frac{p}{q} \in [0,1)$ with prime denominator $q$ written in decimal have a curious property described by Midy's Theorem, namely that two halves of their period (if it is of even length $2n$) sum up to $10^n-1$. A number of results generalise Midy's theorem to expansions of $\frac{p}{q}$ in different integer bases, considering non-prime denominators, or dividing the period into more than two parts. We show that a similar phenomena can be studied even in the context of numeration systems with non-integer bases, as introduced by Rényi. First we define the Midy property for a general real base $β>1$ and derive a necessary condition for validity of the Midy property. For $β=\frac12(1+\sqrt5)$ we characterize prime denominators $q$, which satisfy the property. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_03874 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Midy's Theorem in non-integer bases and divisibility of Fibonacci numbers Masáková, Zuzana Pelantová, Edita Number Theory Combinatorics 11A63, 11B39, 11A07, 11K16 Fractions $\frac{p}{q} \in [0,1)$ with prime denominator $q$ written in decimal have a curious property described by Midy's Theorem, namely that two halves of their period (if it is of even length $2n$) sum up to $10^n-1$. A number of results generalise Midy's theorem to expansions of $\frac{p}{q}$ in different integer bases, considering non-prime denominators, or dividing the period into more than two parts. We show that a similar phenomena can be studied even in the context of numeration systems with non-integer bases, as introduced by Rényi. First we define the Midy property for a general real base $β>1$ and derive a necessary condition for validity of the Midy property. For $β=\frac12(1+\sqrt5)$ we characterize prime denominators $q$, which satisfy the property. |
| title | Midy's Theorem in non-integer bases and divisibility of Fibonacci numbers |
| topic | Number Theory Combinatorics 11A63, 11B39, 11A07, 11K16 |
| url | https://arxiv.org/abs/2401.03874 |