Midy's Theorem in non-integer bases and divisibility of Fibonacci numbers

Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F68407700%3A21340%2F25%3A00377958" target="_blank" >RIV/68407700:21340/25:00377958 - isvavai.cz</a>
Result on the web
<a href="https://doi.org/10.46298/cm.12840" target="_blank" >https://doi.org/10.46298/cm.12840</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.46298/cm.12840" target="_blank" >10.46298/cm.12840</a>

Result language
angličtina
Original language name
Midy's Theorem in non-integer bases and divisibility of Fibonacci numbers
Original language description
Fractions p/qelement[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 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=(1+root 5)/2 we characterize prime denominators q, which satisfy the property.
Czech name
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Czech description
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Type
J<sub>SC</sub> - Article in a specialist periodical, which is included in the SCOPUS database
CEP classification
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OECD FORD branch
10101 - Pure mathematics

Project
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Continuities
I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Publication year
2025
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

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