论文标题
k-将来的佩尔序列中的重新划分
Repdigits in k-generalized Pell sequence
论文作者
论文摘要
令$ k \ geq 2 $，让$（p_ {n}^{（k）}）_ {n \ geq 2-k} $ be $ k $ generalized pell序列由\ begin {equation*} p_ {n}^{（k）} = 2p_ {n-1}^{（k）}+p_ {n-2}^{（k）}+... \ begin {qore*} p _ { - （k-2）}^{（k）} = p _ { - （k-3）}}^{（k）} = \ cdot \ cdot \ cdot = p _ { - 1}^{（k）} = p_ {0}^{（k）} = 0，p_ {1}^{（k）} = 1。在本文中，我们处理diophantine方程\ begin {qore*} p_ {n}^{（k）} = d \ left（\ frac {10^{m} -1} -1} -1} -1} {9} {9} {9} {9} \ right） $ M \ geq 2 $和$ 1 \ leq d \ leq 9 $。我们将在序列$ \ left（p_ {n}^{（k）} \ right）_ {n \ geq 2-k} $中显示至少两个数字的回报是数字\ $ p_ {5}^{5}^{（3）} = 33 $和$ p_和$ p_ {6}}^{6}^{4）} =88。
Let $k\geq 2$ and let $(P_{n}^{(k)})_{n\geq 2-k}$ be $k$-generalized Pell sequence defined by \begin{equation*}P_{n}^{(k)}=2P_{n-1}^{(k)}+P_{n-2}^{(k)}+...+P_{n-k}^{(k)}\end{equation*} for $n\geq 2$ with initial conditions \begin{equation*}P_{-(k-2)}^{(k)}=P_{-(k-3)}^{(k)}=\cdot \cdot \cdot =P_{-1}^{(k)}=P_{0}^{(k)}=0,P_{1}^{(k)}=1. \end{equation*} In this paper, we deal with the Diophantine equation \begin{equation*}P_{n}^{(k)}=d\left( \frac{10^{m}-1}{9}\right)\end{equation*} in positive integers $n,m,k,d$ with $k\geq 2,$ $m\geq 2$ and $1\leq d\leq 9$. We will show that repdigits with at least two digits in the sequence $\left( P_{n}^{(k)}\right)_{n\geq 2-k}$ are the numbers\ $P_{5}^{(3)}=33$ and $P_{6}^{(4)}=88.$
