Cullen numbers and Woodall numbers in generalized Fibonacci sequences

Pub Date : 2024-04-23 DOI:10.1016/j.jnt.2024.03.006

Attila Bérczes , István Pink , Paul Thomas Young

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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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