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

Kód výsledku v 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>

Výsledek na webu

<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>

Jazyk výsledku

angličtina

Název v původním jazyce

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

Popis výsledku v původním jazyce

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.

Název v anglickém jazyce

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

Popis výsledku anglicky

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.

Druh

J_SC - Článek v periodiku v databázi SCOPUS

CEP obor

—

OECD FORD obor

10101 - Pure mathematics

Projekt

—

Návaznosti

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Rok uplatnění

2025

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název periodika

Communications in Mathematics

ISSN

1804-1388

e-ISSN

—

Svazek periodika

33

Číslo periodika v rámci svazku

2

Stát vydavatele periodika

CZ - Česká republika

Počet stran výsledku

17

Strana od-do

—

Kód UT WoS článku

—

EID výsledku v databázi Scopus

2-s2.0-85196186554