{"title":"哪些正整数n可以用n = (x + y + z)(1/x + 1/y + 1/z)的形式表示?","authors":"N. X. Tho","doi":"10.33039/AMI.2021.04.005","DOIUrl":null,"url":null,"abstract":"This paper shows that the equation in the title does not have positive integer solutions when n is divisible by 4. This gives a partial answer to a question by Melvyn Knight. The proof is a mixture of elementary p-adic analysis and elliptic curve theory.","PeriodicalId":43454,"journal":{"name":"Annales Mathematicae et Informaticae","volume":"86 1","pages":""},"PeriodicalIF":0.3000,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"3","resultStr":"{\"title\":\"What positive integers n can be presented in the form n = (x + y + z)(1/x + 1/y + 1/z)?\",\"authors\":\"N. X. Tho\",\"doi\":\"10.33039/AMI.2021.04.005\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"This paper shows that the equation in the title does not have positive integer solutions when n is divisible by 4. This gives a partial answer to a question by Melvyn Knight. The proof is a mixture of elementary p-adic analysis and elliptic curve theory.\",\"PeriodicalId\":43454,\"journal\":{\"name\":\"Annales Mathematicae et Informaticae\",\"volume\":\"86 1\",\"pages\":\"\"},\"PeriodicalIF\":0.3000,\"publicationDate\":\"2021-01-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"3\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Annales Mathematicae et Informaticae\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.33039/AMI.2021.04.005\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Annales Mathematicae et Informaticae","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.33039/AMI.2021.04.005","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS","Score":null,"Total":0}
What positive integers n can be presented in the form n = (x + y + z)(1/x + 1/y + 1/z)?
This paper shows that the equation in the title does not have positive integer solutions when n is divisible by 4. This gives a partial answer to a question by Melvyn Knight. The proof is a mixture of elementary p-adic analysis and elliptic curve theory.
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