On the matricial version of Fermat-Euler congruences
type: Item
instance of: scholarly article
title: On the matricial version of Fermat-Euler congruences
zbMATH Open document ID: 1168.05003
published in: Japanese Journal of Mathematics. 3rd Series
publication date: 2009-02-06T00:00:00Z
review text: The paper discusses some generalizations of the Little Fermat Theorem and its generalization to general \(n\) due to Euler. To be more specific in the two statements \(a^p \equiv a \bmod p\) and \(a^n \equiv a^{n-\varphi(n)} \bmod n\) where \(\varphi(n)\) is the Euler totient function, the \(a\) is replaced by the trace of an integer matrix \(A\). The matricial version of the second statement turns out to be wrong for \(n \neq p^b\) where \(p\) is prime, but seems to hold otherwise. The stated congruences can be translated in particular congruences involving symmetric functions. The congruences on the symmetric functions can be shown to hold if certain congruences between particular multinomial coefficients hold. These congruences are shown to hold (by an inductive argument) provided some basic (starting) conditions hold. These starting conditions have to be verified for every set of parameters separately (infinitely many). In the paper the author verifies starting conditions for \(p=3\) and \(p=5\) and some parameter \(r=3\), and claims that a computer verified them for all primes less than 30. This then also would prove the matricial Euler congruences mod \(p^r\) \(p= 3\) or 5 and \(r\) less than 5. On the fly Arnold introduces what he calls Newton-Girard formulas expressing \(x_1^n+\dots+x_r^n\) in terms of the moments of the elementary symmetric functions \(\sigma_1=x_1+\dots +x_r\), \(\sigma_2=x_1x_2+\dots\), \(\sigma_r=x_1x_2\cdots x_r\), that were already proved in an earlier paper of the same author. For checking the initial conditions on the multinomial coefficients the author found some useful shortcuts reducing the amount of work considerably. The paper does not prove the presented matricial form of the Euler congruences but rather indicates a way to prove it for separate sets of parameters. Finally the paper connects the presented material to some general number theoretic problems like the behaviour of the minimal period of the geometric progression of r…
reviewed by: Herman J. Tiersma
Mathematics Subject Classification ID: 11A07
Mathematics Subject Classification ID: 11B65
Mathematics Subject Classification ID: 11C20
zbMATH DE Number: 5503380
zbMATH Keywords: Newton-Girard formula
zbMATH Keywords: multinomial coefficients
zbMATH Keywords: Cesaro averaging
zbMATH Keywords: symmetric functions
zbMATH Keywords: little Fermat theorem
zbMATH Keywords: Euler congruences
author: Vładimir I. Arnold
MaRDI profile type: Publication
full work available at URL: s11537-006-0501-6
OpenAlex ID: W4255559824
cites work: Empirical estimates of the average orders of orbits period lengths in Euler groups
cites work: Fermat dynamics, matrix arithmetics, finite circles, and finite Lobachevsky planes
cites work: A Coloring Proof of a Generalisation of Fermat's Little Theorem
cites work: Q5788436
DOI: 10.1007/S11537-006-0501-6
description: scientific article; zbMATH DE number 5503380