An extension of Aigner’s theorem

In 1957, Aigner (Monatsh Math 61:147–150, 1957) showed that the equations \(x^6+y^6=z^6\) and \(x^9+y^9=z^9\) have no solutions in any quadratic number field with \(xyz\ne 0\). We show that Aigner’s result holds for all equations \(x^{3n}+y^{3n}=z^{3n}\), where \(n\ge 2\) is a positive integer. The proof combines Aigner’s idea with deep results on Fermat’s equation and its variants.