{"title":"On Pillai's Problem involving Lucas sequences of the second kind.","authors":"Sebastian Heintze, Volker Ziegler","doi":"10.1007/s40993-024-00534-5","DOIUrl":null,"url":null,"abstract":"<p><p>In this paper, we consider the Diophantine equation <math><mrow><msub><mi>V</mi><mi>n</mi></msub><mo>-</mo><msup><mi>b</mi><mi>m</mi></msup><mo>=</mo><mi>c</mi></mrow></math> for given integers <i>b</i>, <i>c</i> with <math><mrow><mi>b</mi><mo>≥</mo><mn>2</mn></mrow></math>, whereas <math><msub><mi>V</mi><mi>n</mi></msub></math> varies among Lucas-Lehmer sequences of the second kind. We prove under some technical conditions that if the considered equation has at least three solutions (<i>n</i>, <i>m</i>) , then there is an upper bound on the size of the solutions as well as on the size of the coefficients in the characteristic polynomial of <math><msub><mi>V</mi><mi>n</mi></msub></math>.</p>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC11090840/pdf/","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1007/s40993-024-00534-5","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"2024/5/13 0:00:00","PubModel":"Epub","JCR":"","JCRName":"","Score":null,"Total":0}
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Abstract
In this paper, we consider the Diophantine equation for given integers b, c with , whereas varies among Lucas-Lehmer sequences of the second kind. We prove under some technical conditions that if the considered equation has at least three solutions (n, m) , then there is an upper bound on the size of the solutions as well as on the size of the coefficients in the characteristic polynomial of .
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