Periods and Young's diagram of Fermat-Euler's geometrical progressions of residues
type: Item
instance of: scholarly article
title: Periods and Young's diagram of Fermat-Euler's geometrical progressions of residues
zbMATH Open document ID: 1221.11005
published in: Functional Analysis and Other Mathematics
publication date: 2010-12-15T00:00:00Z
review text: The paper under review is an excellent paper that is full of ideas for interested researchers in the subject. Let \(n\) be a positive integer and consider the geometric sequence of residues modulo \(n\); \[ \{a^{t}\mid t=1,2,\dots\}\pmod n \] for \(a \in {\mathbb Z}_n\). Such sequence is always periodic starting from some place, \[ a^{t+T(a,n)}\equiv a^t\pmod n \] for every sufficiently large \(t\). What is the minimal period length \(T(a,n)\) of this sequence? The corresponding Young diagram of \(n\) values of \(T(a,n)\), describes the partition of \(n\) residues \(a\) into the classes, for whose sequence the period's length has a fixed value. The author of the paper under review studies the arithmetic and statistical properties of the function \(T(a,n)\) of natural variables \(a\) and \(n\). The author's clever presentation and approach to the problem guides readers to a world of interested questions for further investigations and research.
reviewed by: Manouchehr Misaghian
Mathematics Subject Classification ID: 11A15
Mathematics Subject Classification ID: 11B50
Mathematics Subject Classification ID: 37B50
zbMATH DE Number: 5826919
zbMATH Keywords: small Fermat theorem
zbMATH Keywords: Euler's function
zbMATH Keywords: random permutations cycles
zbMATH Keywords: Cesaro averaging
zbMATH Keywords: Euler's groups
zbMATH Keywords: random divisors statistics
zbMATH Keywords: Gauss integers
zbMATH Keywords: complex prime numbers
author: Vładimir I. Arnold
MaRDI profile type: Publication
full work available at URL: s11853-010-0042-7
OpenAlex ID: W1972663065
cites work: Permutations
DOI: 10.1007/S11853-010-0042-7
description: scientific article; zbMATH DE number 5826919