A natural extension of the Pythagorean equation to higher dimensions
type: Item
instance of: scholarly article
title: A natural extension of the Pythagorean equation to higher dimensions
zbMATH Open document ID: 1188.11010
author: Marc Chamberland
published in: The Ramanujan Journal
publication date: 2008-09-12T00:00:00Z
review text: The equation \(Q_n(x_1,x_2,\dots,x_n)=s^n\), where \(Q_n(x_1,x_2,\dots,x_n)\) denotes the circular matrix resulting from the (first) row \((x_1,x_2,\dots,x_n)\), is studied. The complete rational solution of this equation (when \(s\neq 0\)) is given in terms of \(n+1\) parameters. For example, when \(n=3\), then \(Q_3(x,y,z)=x^3+y^3+z^3-3xyz\), and the complete rational solution to \(Q_3(x,y,z)=s^3\) is given by \[ x=a^3-c^3-t^3\,,\; y=b^3-a^3-t^3\,,\; z=c^3-b^3-t^3\,,s={-3t \over 2}(a^2+b^2+c^2)\,, \] where \(a,b,c,t\) are rational parameters and \(a+b+c=0\). A number of interesting properties of the function \(Q_n\) are proved. The treatment, though elementary, is not straightforward and the exposition is clear. To this reviewer's opinion, however, the connexion of the Diophantine equation \(Q_n(x_1,x_2,\dots,x_n)=s^n\) to the Pythagorean equation \(x^2+y^2=z^2\) is so indirect, that he finds the title of the paper rather unsuccessful.
Mathematics Subject Classification ID: 11D41
Mathematics Subject Classification ID: 11D25
Mathematics Subject Classification ID: 11D99
zbMATH DE Number: 5343450
zbMATH Keywords: circular matrices
zbMATH Keywords: Diophantine equations involving circular matrices
zbMATH Keywords: Pythagorean equation
zbMATH Keywords: theta functions
reviewed by: Nikos Tzanakis
Wikidata QID: Q114223588
MaRDI profile type: Publication
full work available at URL: s11139-007-9107-8
OpenAlex ID: W1994223622
cites work: Q4276250
cites work: Q4853009
cites work: Q3751634
cites work: A Cubic Counterpart of Jacobi's Identity and the AGM
cites work: Some Cubic Modular Identities of Ramanujan
cites work: Q4359658
cites work: An Amazing Identity of Ramanujan
cites work: A Proof in the Spirit of Zeilberger of an Amazing Identity of Ramanujan
cites work: Beyond Sin and Cos
cites work: Cubic and general extensions of the Lorentz transformation
cites work: Q3904661
cites work: Q3999646
cites work: Generalized Hyperbolic Functions
cites work: Q4226695
DOI: 10.1007/S11139-007-9107-8
description: scientific article; zbMATH DE number 5343450