Cullen numbers and Woodall numbers in generalized Fibonacci sequences

IF 0.6 3区数学 Q3 MATHEMATICS

Journal of Number Theory Pub Date : 2024-04-23 DOI:10.1016/j.jnt.2024.03.006

Attila Bérczes , István Pink , Paul Thomas Young

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引用次数: 0

Abstract

Recently Bilu, Marques and Togbé [4] gave a general effective finiteness result on the equation $F_{n}^{(k)} = C_{m},$ where $F_{n}^{(k)}$ denotes the k-generalized Fibonacci-sequence and $C_{m}$ the sequence of Cullen numbers, by giving explicit absolute bounds for $n, k, m$ . However, the authors in [4] explained that their bounds were too large to use Dujella-Pethő reduction to completely solve the equation in question. In the present paper, using the bounds established by Bilu, Marques and Togbé in [4] and a different approach based on 2-adic analysis, we completely solve this equation. Further, using the same approach we also solve the corresponding equation for Woodall numbers.

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广义斐波那契数列中的库伦数和伍德尔数

最近，Bilu、Marques 和 Togbé [4] 通过给出 n、k、m 的明确绝对界限，给出了方程 Fn(k)=Cm 的一般有效有限性结果，其中 Fn(k) 表示 k 个广义的斐波纳契数列，Cm 表示库伦数列。然而，[4] 中的作者解释说，他们的界限太大，无法使用 Dujella-Pethő 还原法完全求解相关方程。在本文中，我们利用 Bilu、Marques 和 Togbé 在 [4] 中建立的边界，以及基于 2-adic 分析的不同方法，完全求解了这个方程。此外，我们还利用同样的方法求解了伍德尔数的相应方程。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Journal of Number Theory 数学-数学

CiteScore

1.30

自引率

14.30%

发文量

122

审稿时长

16 weeks

期刊介绍： The Journal of Number Theory (JNT) features selected research articles that represent the broad spectrum of interest in contemporary number theory and allied areas. A valuable resource for mathematicians, the journal provides an international forum for the publication of original research in this field. The Journal of Number Theory is encouraging submissions of quality, long articles where most or all of the technical details are included. The journal now considers and welcomes also papers in Computational Number Theory. Starting in May 2019, JNT will have a new format with 3 sections: JNT Prime targets (possibly very long with complete proofs) high impact papers. Articles published in this section will be granted 1 year promotional open access. JNT General Section is for shorter papers. We particularly encourage submission from junior researchers. Every attempt will be made to expedite the review process for such submissions. Computational JNT . This section aims to provide a forum to disseminate contributions which make significant use of computer calculations to derive novel number theoretic results. There will be an online repository where supplementary codes and data can be stored.