{"title":"论涉及第二类卢卡斯序列的皮莱问题","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":"{\"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}","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}
引用次数: 0
摘要
在本文中，我们考虑了给定整数 b, c 的二阶方程 Vn-bm=c，b≥2，而 Vn 在第二类卢卡斯-雷默序列中变化。我们在一些技术条件下证明，如果所考虑的方程至少有三个解 (n, m) ，那么解的大小以及 Vn 的特征多项式系数的大小都有一个上限。
On Pillai's Problem involving Lucas sequences of the second kind.
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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