论文标题
复发序列术语的库伦数字
Cullen numbers in sums of terms of recurrence sequence
论文作者
论文摘要
令$（u_n）_ {n \ geq 0} $为固定的线性复发序列的整数至少两个，对于任何积极的整数$ \ ell $，令$ \ ell \ ell \ cdot 2^{\ ell} + 1 $ be cullen编号。最近，在\ cite {bmt}中，已经研究了在某些弱假设下的线性复发序列$（u_n）_ {n \ geq 0} $的广义cullen数字。但是，他们的证明存在错误。在本文中，我们概括了他们的工作，并且结果解决了错误。特别是，对于给定的多项式$ q（x）\ in \ mathbb {z} [x] $，我们考虑使用diophantine方程$ u_ {n_1} + \ cdots + u_ {n_k} = \ el \ ell \ ell \ ell \ cdot x^{\ ell} {\ ell} + q（x）$，以及prove prove anitys效果。此外，我们以示例演示了我们的方法。
Let $(U_n)_{n\geq 0}$ be a fixed linear recurrence sequence of integers with order at least two, and for any positive integer $\ell$, let $\ell \cdot 2^{\ell} + 1$ be a Cullen number. Recently in \cite{bmt}, generalized Cullen numbers in terms of linear recurrence sequence $(U_n)_{n\geq 0}$ under certain weak assumptions has been studied. However, there is an error in their proof. In this paper, we generalize their work, as well as our result fixes their error. In particular, for a given polynomial $Q(x) \in \mathbb{Z}[x]$ we consider the Diophantine equation $U_{n_1} + \cdots + U_{n_k} = \ell \cdot x^{\ell} + Q(x)$, and prove effective finiteness result. Furthermore, we demonstrate our method by an example.
