{"title":"On the $x$--coordinates of Pell equations that are products of two Lucas numbers","authors":"Mahadi Ddamulira","doi":"10.33774/COE-2020-27J3Q","DOIUrl":"https://doi.org/10.33774/COE-2020-27J3Q","url":null,"abstract":"Let $ {L_n}_{nge 0} $ be the sequence of Lucas numbers given by $ L_0=2, ~ L_1=1 $ and $ L_{n+2}=L_{n+1}+L_n $ for all $ nge 0 $. In this paper, for an integer $dgeq 2$ which is square-free, we show that there is at most one value of the positive integer $x$ participating in the Pell equation $x^{2}-dy^{2}=pm 1$ which is a product of two Lucas numbers, with a few exceptions that we completely characterize.","PeriodicalId":47144,"journal":{"name":"FIBONACCI QUARTERLY","volume":"28 1","pages":"18-37"},"PeriodicalIF":0.6,"publicationDate":"2020-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83718287","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}