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Midy's Theorem in non-integer bases and divisibility of Fibonacci numbers
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 $\fr...
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| Published in: | Communications in Mathematics 2024-05, Vol.33 (2025), Issue 2... |
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| Format: | Article |
| Language: | English |
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| container_title | Communications in Mathematics |
| container_volume | 33 (2025), Issue 2... |
| creator | Masáková, Zuzana Pelantová, Edita |
| description | 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 R\'enyi. First we define the Midy property for a general real
base $\beta >1$ and derive a necessary condition for validity of the Midy
property. For $\beta =\frac12(1+\sqrt5)$ we characterize prime denominators
$q$, which satisfy the property. |
| doi_str_mv | 10.46298/cm.12840 |
| format | article |
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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 R\'enyi. First we define the Midy property for a general real
base $\beta >1$ and derive a necessary condition for validity of the Midy
property. For $\beta =\frac12(1+\sqrt5)$ we characterize prime denominators
$q$, which satisfy the property.</description><identifier>ISSN: 2336-1298</identifier><identifier>EISSN: 2336-1298</identifier><identifier>DOI: 10.46298/cm.12840</identifier><language>eng</language><ispartof>Communications in Mathematics, 2024-05, Vol.33 (2025), Issue 2...</ispartof><lds50>peer_reviewed</lds50><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><link.rule.ids>314,777,781,27905,27906</link.rule.ids></links><search><creatorcontrib>Masáková, Zuzana</creatorcontrib><creatorcontrib>Pelantová, Edita</creatorcontrib><title>Midy's Theorem in non-integer bases and divisibility of Fibonacci numbers</title><title>Communications in Mathematics</title><description>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 R\'enyi. First we define the Midy property for a general real
base $\beta >1$ and derive a necessary condition for validity of the Midy
property. For $\beta =\frac12(1+\sqrt5)$ we characterize prime denominators
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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 R\'enyi. First we define the Midy property for a general real
base $\beta >1$ and derive a necessary condition for validity of the Midy
property. For $\beta =\frac12(1+\sqrt5)$ we characterize prime denominators
$q$, which satisfy the property.</abstract><doi>10.46298/cm.12840</doi><oa>free_for_read</oa></addata></record> |
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| ispartof | Communications in Mathematics, 2024-05, Vol.33 (2025), Issue 2... |
| issn | 2336-1298 2336-1298 |
| language | eng |
| recordid | cdi_crossref_primary_10_46298_cm_12840 |
| source | Freely Accessible Science Journals - check A-Z of ejournals |
| title | Midy's Theorem in non-integer bases and divisibility of Fibonacci numbers |
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