{"title":"斐波那契数是斐波那契数和卢卡斯数的混合并集","authors":"Alaa Altassan, Murat Alan","doi":"10.1515/ms-2024-0042","DOIUrl":null,"url":null,"abstract":"Let (<jats:italic>F<jats:sub>n</jats:sub> </jats:italic>)<jats:sub> <jats:italic>n</jats:italic>≥0</jats:sub> and (<jats:italic>L<jats:sub>n</jats:sub> </jats:italic>)<jats:sub> <jats:italic>n</jats:italic>≥0</jats:sub> be the Fibonacci and Lucas sequences, respectively. In this paper we determine all Fibonacci numbers which are mixed concatenations of a Fibonacci and a Lucas numbers. By mixed concatenations of <jats:italic>a</jats:italic> and <jats:italic>b</jats:italic>, we mean the both concatenations <jats:overline> <jats:italic>ab</jats:italic> </jats:overline> and <jats:overline> <jats:italic>ba</jats:italic> </jats:overline> together, where <jats:italic>a</jats:italic> and <jats:italic>b</jats:italic> are any two nonnegative integers. So, the mathematical formulation of this problem leads us searching the solutions of two Diophantine equations <jats:italic>F<jats:sub>n</jats:sub> </jats:italic> = 10<jats:sup> <jats:italic>d</jats:italic> </jats:sup> <jats:italic>F<jats:sub>m</jats:sub> </jats:italic> + <jats:italic>L<jats:sub>k</jats:sub> </jats:italic> and <jats:italic>F<jats:sub>n</jats:sub> </jats:italic> = 10<jats:sup> <jats:italic>d</jats:italic> </jats:sup> <jats:italic>L<jats:sub>m</jats:sub> </jats:italic> + <jats:italic>F<jats:sub>k</jats:sub> </jats:italic> in nonnegative integers (<jats:italic>n</jats:italic>, <jats:italic>m</jats:italic>, <jats:italic>k</jats:italic>), where <jats:italic>d</jats:italic> denotes the number of digits of <jats:italic>L<jats:sub>k</jats:sub> </jats:italic> and <jats:italic>F<jats:sub>k</jats:sub> </jats:italic>, respectively. We use lower bounds for linear forms in logarithms and reduction method in Diophantine approximation to get the results.","PeriodicalId":18282,"journal":{"name":"Mathematica Slovaca","volume":null,"pages":null},"PeriodicalIF":0.9000,"publicationDate":"2024-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Fibonacci numbers as mixed concatenations of Fibonacci and Lucas numbers\",\"authors\":\"Alaa Altassan, Murat Alan\",\"doi\":\"10.1515/ms-2024-0042\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Let (<jats:italic>F<jats:sub>n</jats:sub> </jats:italic>)<jats:sub> <jats:italic>n</jats:italic>≥0</jats:sub> and (<jats:italic>L<jats:sub>n</jats:sub> </jats:italic>)<jats:sub> <jats:italic>n</jats:italic>≥0</jats:sub> be the Fibonacci and Lucas sequences, respectively. In this paper we determine all Fibonacci numbers which are mixed concatenations of a Fibonacci and a Lucas numbers. By mixed concatenations of <jats:italic>a</jats:italic> and <jats:italic>b</jats:italic>, we mean the both concatenations <jats:overline> <jats:italic>ab</jats:italic> </jats:overline> and <jats:overline> <jats:italic>ba</jats:italic> </jats:overline> together, where <jats:italic>a</jats:italic> and <jats:italic>b</jats:italic> are any two nonnegative integers. So, the mathematical formulation of this problem leads us searching the solutions of two Diophantine equations <jats:italic>F<jats:sub>n</jats:sub> </jats:italic> = 10<jats:sup> <jats:italic>d</jats:italic> </jats:sup> <jats:italic>F<jats:sub>m</jats:sub> </jats:italic> + <jats:italic>L<jats:sub>k</jats:sub> </jats:italic> and <jats:italic>F<jats:sub>n</jats:sub> </jats:italic> = 10<jats:sup> <jats:italic>d</jats:italic> </jats:sup> <jats:italic>L<jats:sub>m</jats:sub> </jats:italic> + <jats:italic>F<jats:sub>k</jats:sub> </jats:italic> in nonnegative integers (<jats:italic>n</jats:italic>, <jats:italic>m</jats:italic>, <jats:italic>k</jats:italic>), where <jats:italic>d</jats:italic> denotes the number of digits of <jats:italic>L<jats:sub>k</jats:sub> </jats:italic> and <jats:italic>F<jats:sub>k</jats:sub> </jats:italic>, respectively. We use lower bounds for linear forms in logarithms and reduction method in Diophantine approximation to get the results.\",\"PeriodicalId\":18282,\"journal\":{\"name\":\"Mathematica Slovaca\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.9000,\"publicationDate\":\"2024-07-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Mathematica Slovaca\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1515/ms-2024-0042\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematica Slovaca","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1515/ms-2024-0042","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
摘要
设 (Fn ) n≥0 和 (Ln ) n≥0 分别为斐波那契数列和卢卡斯数列。在本文中，我们确定所有斐波那契数都是一个斐波那契数和一个卢卡斯数的混合并集。我们所说的 a 和 b 的混合并集是指 ab 和 ba 的并集，其中 a 和 b 是任意两个非负整数。因此，这个问题的数学表达式引导我们在非负整数 (n, m, k) 中搜索两个二叉方程 Fn = 10 d Fm + Lk 和 Fn = 10 d Lm + Fk 的解，其中 d 分别表示 Lk 和 Fk 的位数。我们利用对数线性形式的下界和 Diophantine 近似中的还原法得出结果。
Fibonacci numbers as mixed concatenations of Fibonacci and Lucas numbers
Let (Fn)n≥0 and (Ln)n≥0 be the Fibonacci and Lucas sequences, respectively. In this paper we determine all Fibonacci numbers which are mixed concatenations of a Fibonacci and a Lucas numbers. By mixed concatenations of a and b, we mean the both concatenations ab and ba together, where a and b are any two nonnegative integers. So, the mathematical formulation of this problem leads us searching the solutions of two Diophantine equations Fn = 10dFm + Lk and Fn = 10dLm + Fk in nonnegative integers (n, m, k), where d denotes the number of digits of Lk and Fk, respectively. We use lower bounds for linear forms in logarithms and reduction method in Diophantine approximation to get the results.
期刊介绍：
Mathematica Slovaca, the oldest and best mathematical journal in Slovakia, was founded in 1951 at the Mathematical Institute of the Slovak Academy of Science, Bratislava. It covers practically all mathematical areas. As a respectful international mathematical journal, it publishes only highly nontrivial original articles with complete proofs by assuring a high quality reviewing process. Its reputation was approved by many outstanding mathematicians who already contributed to Math. Slovaca. It makes bridges among mathematics, physics, soft computing, cryptography, biology, economy, measuring, etc. The Journal publishes original articles with complete proofs. Besides short notes the journal publishes also surveys as well as some issues are focusing on a theme of current interest.
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