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An extension of Aigner’s theorem

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Abstract

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.

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Authors and Affiliations

  1. Hanoi University of Science and Technology, Hanoi, Vietnam

    Nguyen Xuan Tho

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  1. Nguyen Xuan Tho
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Correspondence to Nguyen Xuan Tho.

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Communicated by Alberto Minguez.

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Tho, N.X. An extension of Aigner’s theorem. Monatsh Math 204, 191–195 (2024). https://doi.org/10.1007/s00605-023-01913-3

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