{"title":"一对具有平衡数和Lucas平衡数互质系数的线性双变量丢番图方程的解","authors":"R. K. Davala","doi":"10.7546/nntdm.2023.29.3.495-502","DOIUrl":null,"url":null,"abstract":"Let $B_n$ and $C_n$ be the $n$-th balancing and Lucas-balancing numbers, respectively. We consider the Diophantine equations $ax+by=\\frac{1}{2}(a-1)(b-1)$ and $1+ax+by=\\frac{1}{2}(a-1)(b-1)$ for $(a,b)$ $\\in$ $ \\{(B_n,B_{n+1}),(B_{2n-1},B_{2n+1}), (B_n,C_n),(C_n,C_{n+1})\\}$ and present the non-negative integer solutions of the Diophantine equations in each case.","PeriodicalId":44060,"journal":{"name":"Notes on Number Theory and Discrete Mathematics","volume":" ","pages":""},"PeriodicalIF":0.4000,"publicationDate":"2023-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":"{\"title\":\"Solution to a pair of linear, two-variable, Diophantine equations with coprime coefficients from balancing and Lucas-balancing numbers\",\"authors\":\"R. K. Davala\",\"doi\":\"10.7546/nntdm.2023.29.3.495-502\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Let $B_n$ and $C_n$ be the $n$-th balancing and Lucas-balancing numbers, respectively. We consider the Diophantine equations $ax+by=\\\\frac{1}{2}(a-1)(b-1)$ and $1+ax+by=\\\\frac{1}{2}(a-1)(b-1)$ for $(a,b)$ $\\\\in$ $ \\\\{(B_n,B_{n+1}),(B_{2n-1},B_{2n+1}), (B_n,C_n),(C_n,C_{n+1})\\\\}$ and present the non-negative integer solutions of the Diophantine equations in each case.\",\"PeriodicalId\":44060,\"journal\":{\"name\":\"Notes on Number Theory and Discrete Mathematics\",\"volume\":\" \",\"pages\":\"\"},\"PeriodicalIF\":0.4000,\"publicationDate\":\"2023-07-15\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"1\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Notes on Number Theory and Discrete Mathematics\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.7546/nntdm.2023.29.3.495-502\",\"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":"Notes on Number Theory and Discrete Mathematics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.7546/nntdm.2023.29.3.495-502","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS","Score":null,"Total":0}
Solution to a pair of linear, two-variable, Diophantine equations with coprime coefficients from balancing and Lucas-balancing numbers
Let $B_n$ and $C_n$ be the $n$-th balancing and Lucas-balancing numbers, respectively. We consider the Diophantine equations $ax+by=\frac{1}{2}(a-1)(b-1)$ and $1+ax+by=\frac{1}{2}(a-1)(b-1)$ for $(a,b)$ $\in$ $ \{(B_n,B_{n+1}),(B_{2n-1},B_{2n+1}), (B_n,C_n),(C_n,C_{n+1})\}$ and present the non-negative integer solutions of the Diophantine equations in each case.
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