A natural extension of the Pythagorean equation to higher dimensions
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.
Diophantine equations involving circular matrices
scientific article; zbMATH DE number 5343450