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引用次数: 0
摘要
摘要 设 k ≥ 2。众所周知的佩尔序列的广义化是 k-Pell 序列，其前 k 项为 0，...，0，1，之后的每项由线性递推公式 pn(k)=2Pn-1(k)+Pn-2(k)+⋯+Pn-k(k)给出。本文的目的是证明 11、33、55、88 和 99 都是可以用两个 k-Pell 的和或差来表示的重数字。我们主要定理的证明使用了代数数对数线性形式的下界，以及改进版的贝克-达文波特还原法（由杜杰拉和佩索提出）。这扩展了布拉沃和埃雷拉 [Repdigits in generalized Pell sequences, Arch.(Brno) 56(4) (2020), 249-262].
On Repdigits Which are Sums or Differences of Two k-Pell Numbers
ABSTRACT Let k ≥ 2. A generalization of the well-known Pell sequence is the k-Pell sequence whose first k terms are 0,…, 0, 1 and each term afterwards is given by the linear recurrence pn(k)=2Pn−1(k)+Pn−2(k)+⋯+Pn−k(k). The goal of this paper is to show that 11, 33, 55, 88 and 99 are all repdigits expressible as sum or difference of two k-Pell. The proof of our main theorem uses lower bounds for linear forms in logarithms of algebraic numbers and a modified version of Baker-Davenport reduction method (due to Dujella and Pethő). This extends a result of Bravo and Herrera [Repdigits in generalized Pell sequences, Arch. Math. (Brno) 56(4) (2020), 249–262].
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