关于线性递推序列的素数幂。

IF 0.5 Q3 MATHEMATICS Annales Mathematiques du Quebec Pub Date : 2021-04-18 DOI:10.1007/s40316-021-00163-9

Japhet Odjoumani, Volker Ziegler

{"title":"关于线性递推序列的素数幂。","authors":"Japhet Odjoumani, Volker Ziegler","doi":"10.1007/s40316-021-00163-9","DOIUrl":null,"url":null,"abstract":"<div>In this paper we consider the Diophantine equation \\(U_n=p^x\\) where \\(U_n\\) is a linear recurrence sequence, p is a prime number, and x is a positive integer. Under some technical hypotheses on \\(U_n\\), we show that, for any p outside of an effectively computable finite set of prime numbers, there exists at most one solution (n, x) to that Diophantine equation. We compute this exceptional set for the Tribonacci sequence and for the Lucas sequence plus one.</div>","PeriodicalId":42753,"journal":{"name":"Annales Mathematiques du Quebec","volume":"47 2","pages":"349 - 366"},"PeriodicalIF":0.5000,"publicationDate":"2021-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s40316-021-00163-9","citationCount":"1","resultStr":"{\"title\":\"On prime powers in linear recurrence sequences\",\"authors\":\"Japhet Odjoumani, Volker Ziegler\",\"doi\":\"10.1007/s40316-021-00163-9\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div>In this paper we consider the Diophantine equation \\\\(U_n=p^x\\\\) where \\\\(U_n\\\\) is a linear recurrence sequence, p is a prime number, and x is a positive integer. Under some technical hypotheses on \\\\(U_n\\\\), we show that, for any p outside of an effectively computable finite set of prime numbers, there exists at most one solution (n, x) to that Diophantine equation. We compute this exceptional set for the Tribonacci sequence and for the Lucas sequence plus one.</div>\",\"PeriodicalId\":42753,\"journal\":{\"name\":\"Annales Mathematiques du Quebec\",\"volume\":\"47 2\",\"pages\":\"349 - 366\"},\"PeriodicalIF\":0.5000,\"publicationDate\":\"2021-04-18\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://sci-hub-pdf.com/10.1007/s40316-021-00163-9\",\"citationCount\":\"1\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Annales Mathematiques du Quebec\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://link.springer.com/article/10.1007/s40316-021-00163-9\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Annales Mathematiques du Quebec","FirstCategoryId":"1085","ListUrlMain":"https://link.springer.com/article/10.1007/s40316-021-00163-9","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}

引用次数: 1

摘要

本文考虑丢番图方程Un=px，其中Un是线性递推序列，p是素数，x是正整数。在Un的一些技术假设下，我们证明，对于有效可计算的有限素数集之外的任何p，该丢番图方程最多存在一个解（n，x）。我们为Tribonacci序列和Lucas序列加1计算这个例外集。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

查看原文

微信好友朋友圈 QQ好友复制链接

本刊更多论文

On prime powers in linear recurrence sequences

In this paper we consider the Diophantine equation \(U_n=p^x\) where \(U_n\) is a linear recurrence sequence, p is a prime number, and x is a positive integer. Under some technical hypotheses on \(U_n\), we show that, for any p outside of an effectively computable finite set of prime numbers, there exists at most one solution (n, x) to that Diophantine equation. We compute this exceptional set for the Tribonacci sequence and for the Lucas sequence plus one.

求助全文

通过发布文献求助，成功后即可免费获取论文全文。去求助

来源期刊

Annales Mathematiques du Quebec MATHEMATICS-

CiteScore

1.10

自引率

0.00%

发文量

期刊介绍： The goal of the Annales mathématiques du Québec (formerly: Annales des sciences mathématiques du Québec) is to be a high level journal publishing articles in all areas of pure mathematics, and sometimes in related fields such as applied mathematics, mathematical physics and computer science. Papers written in French or English may be submitted to one of the editors, and each published paper will appear with a short abstract in both languages. History: The journal was founded in 1977 as „Annales des sciences mathématiques du Québec”, in 2013 it became a Springer journal under the name of “Annales mathématiques du Québec”. From 1977 to 2018, the editors-in-chief have respectively been S. Dubuc, R. Cléroux, G. Labelle, I. Assem, C. Levesque, D. Jakobson, O. Cornea. Les Annales mathématiques du Québec (anciennement, les Annales des sciences mathématiques du Québec) se veulent un journal de haut calibre publiant des travaux dans toutes les sphères des mathématiques pures, et parfois dans des domaines connexes tels les mathématiques appliquées, la physique mathématique et l''informatique. On peut soumettre ses articles en français ou en anglais à l''éditeur de son choix, et les articles acceptés seront publiés avec un résumé court dans les deux langues. Histoire: La revue québécoise “Annales des sciences mathématiques du Québec” était fondée en 1977 et est devenue en 2013 une revue de Springer sous le nom Annales mathématiques du Québec. De 1977 à 2018, les éditeurs en chef ont respectivement été S. Dubuc, R. Cléroux, G. Labelle, I. Assem, C. Levesque, D. Jakobson, O. Cornea.

期刊最新文献

Thin Monodromy in \(\textrm{O}(5)\) Some stability results of positive mass theorem for uniformly asymptotically flat 3-manifolds Sharp upper bounds for Steklov eigenvalues of a hypersurface of revolution with two boundary components in Euclidean space Growth rates of Laplace eigenfunctions on the unit disk On the group of \(\omega ^{k}\)-preserving diffeomorphisms