Fundamental solutions of the equation \(x_ 1^ 2+\dots+x_ n^ 2=mx_ 1\dots x_ n\)
Starting with \textit{A. Hurwitz} [Arch. Math. Phys., III. Ser. 11, 185-196 (1907; JFM 38.0246.01)] the Diophantine equation \[ x_ 1^ 2+ \cdots + x_ n^ 2 = mx_ 1 \dots, x_ n, \quad n \geq 3,\;m \in \mathbb N, \tag{*} \] was investigated many times over the years. In particular, there was a great interest in the existence of solutions in natural numbers. If \((u_ 1, \dots, u_ n) \in \mathbb N^ n\) is a solution of \((*)\) then, for any \(k \in \{1, \dots, n\}\), also \((u_ 1, \dots, u_{k-1}, u_ k', u_{k+1}, \dots,u_ n)\) is a solution where \(u_ k' = mu_ 1 \dots u_{k-1} u_{k+1} \dots u_ n - u_ k\), and such a solution is called adjacent to \((u_ 1, \dots, u_ n)\). A solution \((u_ 1, \dots, u_ n) \in \mathbb N^ n\) of \((*)\) is called fundamental if it has no adjacent solution with lesser height (the height of a solution \((u_ 1, \dots, u_ n) \in \mathbb N^ n\) of \((*)\) is the sum \(u_ 1 + \cdots + u_ n)\). If \(m>{n + 23 \over 12}\) then a fundamental solution \((u_ 1, \dots, u_ n) \in \mathbb N^ n\) of \((*)\) with \(u_ 1 \geq u_ 2 \geq \cdots \geq u_ n\) may be only of the form \((2,2,2,1,\dots,1)\), \((3,3,1,\dots,1)\), \((u,2,1,\dots,1)\), \(u\in \mathbb N\) and \((v,1, \dots,1)\), \(v \in \mathbb N\).
scientific article; zbMATH DE number 578702