On the number of solutions of the equation \(x_ 1^ 2+ \dots+ x_ n^ 2=nx_ 1 \dots x_ n\), not exceeding a given limit (Q1907410)

Let \({\mathfrak M}_n\) be the set of solutions \((u_1, u_2, \dots, u_n)\) of \(x^2_1+ x^2_2+ \dots+ x^2_n= nx_1 x_2 \dots x_n\), \(n\geq 3\), satisfying \(u_1\geq \dots \geq u_n\geq 1\) and \(M_n (x)= \#\{ (u_1, \dots, u_n)\in {\mathfrak M}_n\mid u\leq x\}\). Then for \(x>1\) the inequalities \(C_1 (n) \ln^2 x< M_n(x)< C_2 (n) \ln^{n-1} x\) are valid where \(C_1 (n)\) and \(C_2 (n)\) are positive constants.