{"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":43826,"journal":{"name":"Research in Number Theory","volume":null,"pages":null},"PeriodicalIF":0.6000,"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\":43826,\"journal\":{\"name\":\"Research in Number Theory\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.6000,\"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\":\"Research in Number Theory\",\"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\":\"Q3\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Research in Number Theory","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":"Q3","JCRName":"MATHEMATICS","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 .
期刊介绍：
Research in Number Theory is an international, peer-reviewed Hybrid Journal covering the scope of the mathematical disciplines of Number Theory and Arithmetic Geometry. The Mission of the Journal is to publish high-quality original articles that make a significant contribution to these research areas. It will also publish shorter research communications (Letters) covering nascent research in some of the burgeoning areas of number theory research. This journal publishes the highest quality papers in all of the traditional areas of number theory research, and it actively seeks to publish seminal papers in the most emerging and interdisciplinary areas here as well. Research in Number Theory also publishes comprehensive reviews.
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