A. Hurwitz proposed the
problem of finding all the positive integers z,x = (x1,⋯,xn) satisfying the
diophantine equation x12+⋯+ xn2= z ⋅ x1,⋯,xn. This paper investigates the
question of which values of z can occur, using only the most elementary
techniques. An algorithm is given for determining all permissible values of (z,n)
for all n below a given bound. As an application it is established that the
only possible values in the range z ≧ (n + 15)∕4 are z = n,z = (n + 8)∕3
when n is odd, and z = (n + 15)∕4. As another application the fifteen values
of n ≦ 131,020 for which the only permissible value of z is n have been
found.
