On prime powers in linear recurrence sequences

In this paper we consider the Diophantine equation [Formula: see text] where [Formula: see text] is a linear recurrence sequence, p is a prime number, and x is a positive integer. Under some technical hypotheses on [Formula: see text] , we show that, for any p outside of an effectively computable fi...

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Detalles Bibliográficos
Autores principales: Odjoumani, Japhet, Ziegler, Volker
Formato: Online Artículo Texto
Lenguaje:English
Publicado: Springer International Publishing 2021
Materias:
Article
Acceso en línea:https://www.ncbi.nlm.nih.gov/pmc/articles/PMC10540468/
https://www.ncbi.nlm.nih.gov/pubmed/37780137
http://dx.doi.org/10.1007/s40316-021-00163-9
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author Odjoumani, Japhet
Ziegler, Volker
author_facet Odjoumani, Japhet
Ziegler, Volker
author_sort Odjoumani, Japhet
collection PubMed
description In this paper we consider the Diophantine equation [Formula: see text] where [Formula: see text] is a linear recurrence sequence, p is a prime number, and x is a positive integer. Under some technical hypotheses on [Formula: see text] , 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.
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spelling pubmed-105404682023-09-30 On prime powers in linear recurrence sequences Odjoumani, Japhet Ziegler, Volker Ann Math Quebec Article In this paper we consider the Diophantine equation [Formula: see text] where [Formula: see text] is a linear recurrence sequence, p is a prime number, and x is a positive integer. Under some technical hypotheses on [Formula: see text] , 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. Springer International Publishing 2021-04-18 2023 /pmc/articles/PMC10540468/ /pubmed/37780137 http://dx.doi.org/10.1007/s40316-021-00163-9 Text en © The Author(s) 2021 https://creativecommons.org/licenses/by/4.0/Open AccessThis article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/ (https://creativecommons.org/licenses/by/4.0/) .
spellingShingle Article
Odjoumani, Japhet
Ziegler, Volker
On prime powers in linear recurrence sequences
title On prime powers in linear recurrence sequences
title_full On prime powers in linear recurrence sequences
title_fullStr On prime powers in linear recurrence sequences
title_full_unstemmed On prime powers in linear recurrence sequences
title_short On prime powers in linear recurrence sequences
title_sort on prime powers in linear recurrence sequences
topic Article
url https://www.ncbi.nlm.nih.gov/pmc/articles/PMC10540468/
https://www.ncbi.nlm.nih.gov/pubmed/37780137
http://dx.doi.org/10.1007/s40316-021-00163-9
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