{"title":"Multiplicative Dependent Pairs in the Sequence of Padovan Numbers","authors":"Mitashree Behera, Prasanta Kumar Ray","doi":"10.1515/ms-2023-0083","DOIUrl":null,"url":null,"abstract":"ABSTRACT The Padovan sequence { P n } n ≥0 is a ternary recurrent sequence defined recursively by the relation P n = P n –2 + P n –3 with initials P 0 = P 1 = P 2 = 1. In this note, we search all pairs of multiplicative dependent vectors whose coordinates are Padovan numbers. For this purpose, we apply Matveev’s theorem to find the lower bounds of the non-zero linear forms in logarithms. Techniques involving the LLL algorithm and the theory of continued fraction are utilized to reduce the bounds.","PeriodicalId":18282,"journal":{"name":"Mathematica Slovaca","volume":"60 1","pages":"0"},"PeriodicalIF":0.9000,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematica Slovaca","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1515/ms-2023-0083","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
Abstract
ABSTRACT The Padovan sequence { P n } n ≥0 is a ternary recurrent sequence defined recursively by the relation P n = P n –2 + P n –3 with initials P 0 = P 1 = P 2 = 1. In this note, we search all pairs of multiplicative dependent vectors whose coordinates are Padovan numbers. For this purpose, we apply Matveev’s theorem to find the lower bounds of the non-zero linear forms in logarithms. Techniques involving the LLL algorithm and the theory of continued fraction are utilized to reduce the bounds.
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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