A unified way for obtaining dividing formulas \(n| Q(n)\)
type: Item
instance of: scholarly article
title: A unified way for obtaining dividing formulas \(n| Q(n)\)
zbMATH Open document ID: 0922.11015
author: Chyi-Lung Lin
published in: Taiwanese Journal of Mathematics
publication date: 1999-10-17T00:00:00Z
review text: Using a function \(f(x)= f^{[1]}(x)\) and defining \(f^{[n]}(x)= f(f^{[n- 1]}(x))\), \(n= 2,3,\dots\) the period-\(n\) points of \(f\) are given by \(f^{[n]}(x)= x\) and \(f^{[i]}(x)\neq x\) for \(i= 2,\dots, n-1\). It is shown that \(n\mid N(n)\), where \(N(n)\) denotes the number of period-\(n\) points of \(f\). By this general result, special dividing formulas, e.g. (1) \(n\mid a^n- a\) for \(n\) prime and \(a= 2,3,\dots\), (2) \(n\mid a^{\varphi(n)- 1}\), \((a,n)= 1\), where \(\varphi(n)\) is Euler's \(\varphi\)-function, (3) \(n\mid (F_{n+ 1}+ F_{n- 1}- 1)\), \(n\) prime and \(F_n\) is the \(n\)th Fibonacci number can be derived.
Mathematics Subject Classification ID: 11B83
Mathematics Subject Classification ID: 11A07
Mathematics Subject Classification ID: 11B39
zbMATH DE Number: 1269916
zbMATH Keywords: number of period-\(n\) points
zbMATH Keywords: dividing formulas
zbMATH Keywords: Fibonacci number
reviewed by: Joachim Piehler
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.11650/twjm/1500407018
OpenAlex ID: W4236994580
DOI: 10.11650/TWJM/1500407018
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description: scientific article; zbMATH DE number 1269916