{"title":"On the x-coordinates of Pell equations that are products of two Pell numbers","authors":"Mahadi Ddamulira, Florian Luca","doi":"10.1515/ms-2024-0004","DOIUrl":null,"url":null,"abstract":"Let (<jats:italic>P<jats:sub>m</jats:sub> </jats:italic>)<jats:sub> <jats:italic>m</jats:italic>≥0</jats:sub> be the sequence of Pell numbers given by <jats:italic>P</jats:italic> <jats:sub>0</jats:sub> = 0, <jats:italic>P</jats:italic> <jats:sub>1</jats:sub> = 1, and <jats:italic>P</jats:italic> <jats:sub> <jats:italic>m</jats:italic>+2</jats:sub> = 2<jats:italic>P</jats:italic> <jats:sub> <jats:italic>m</jats:italic>+1</jats:sub> + <jats:italic>P<jats:sub>m</jats:sub> </jats:italic> for all <jats:italic>m</jats:italic> ≥ 0. In this paper, for an integer <jats:italic>d</jats:italic> ≥ 2 which is square free, we show that there is at most one value of the positive integer <jats:italic>x</jats:italic> participating in the Pell equation <jats:italic>x</jats:italic> <jats:sup>2</jats:sup> − <jats:italic>dy</jats:italic> <jats:sup>2</jats:sup> = ± 1, which is a product of two Pell numbers.","PeriodicalId":18282,"journal":{"name":"Mathematica Slovaca","volume":"32 1","pages":""},"PeriodicalIF":0.9000,"publicationDate":"2024-05-28","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-2024-0004","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
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Abstract
Let (Pm)m≥0 be the sequence of Pell numbers given by P0 = 0, P1 = 1, and Pm+2 = 2Pm+1 + Pm for all m ≥ 0. In this paper, for an integer d ≥ 2 which is square free, we show that there is at most one value of the positive integer x participating in the Pell equation x2 − dy2 = ± 1, which is a product of two Pell numbers.
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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