{"title":"An extension of Aigner’s theorem","authors":"Nguyen Xuan Tho","doi":"10.1007/s00605-023-01913-3","DOIUrl":null,"url":null,"abstract":"<p>In 1957, Aigner (Monatsh Math 61:147–150, 1957) showed that the equations <span>\\(x^6+y^6=z^6\\)</span> and <span>\\(x^9+y^9=z^9\\)</span> have no solutions in any quadratic number field with <span>\\(xyz\\ne 0\\)</span>. We show that Aigner’s result holds for all equations <span>\\(x^{3n}+y^{3n}=z^{3n}\\)</span>, where <span>\\(n\\ge 2\\)</span> is a positive integer. The proof combines Aigner’s idea with deep results on Fermat’s equation and its variants.</p>","PeriodicalId":18913,"journal":{"name":"Monatshefte für Mathematik","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2023-11-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Monatshefte für Mathematik","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1007/s00605-023-01913-3","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
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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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