Kaplansky's ternary quadratic form (Q5931274)

A nonnegative integer \(N\) is called eligible with respect to a form \(f\) if there are no modularity conditions preventing \(f\) from representing \(N\). Now let \(f= x^2+y^2+ 7z^2\) (Kaplansky's ternary quadratic form). It is proved that if \(N\) is a nonnegative eligible integer for \(f\), which is coprime to 7 and not represented by \(f\), then \(N\) is square-free. (In the proof there is a mistake: \(A_2\) has 4 automorphs (instead of 2). But the result is correct).