{"title":"艾格纳定理的推广","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":"5 4","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2023-11-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"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\":\"5 4\",\"pages\":\"\"},\"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}","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}
引用次数: 0
摘要
1957年，Aigner (Monatsh Math 61:147-150, 1957)证明了方程\(x^6+y^6=z^6\)和\(x^9+y^9=z^9\)在含有\(xyz\ne 0\)的任何二次数域中都无解。我们证明Aigner的结果适用于所有方程\(x^{3n}+y^{3n}=z^{3n}\)，其中\(n\ge 2\)是一个正整数。这个证明结合了艾格纳的思想和费马方程及其变体的深刻结果。
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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