论文标题
$ p $ - 亚种商集：线性复发序列
$p$-Adic quotient sets: linear recurrence sequences
论文作者
论文摘要
令$（x_n）_ {n \ geq0} $为订单$ k \ geq2 $满足$$ x_n = a_1x_ {n-1}+a_2x_ {n-2}+dots+dots+a_kx_ { $ a_1，\ dots，a_k，x_0，\ dots，x_ {k-1} \ in \ mathbb {z}，$ a_k \ neq0 $。在[$ k $ generalise fibonacci数字的商集中，在$ \ mathbb {q} _p $'，\ emph {bull中密集。奥斯特。数学。 Soc。} \ TextBf {96}（2017），24-29]，Sanna提出了一个开放的问题，以对Primes $ P $进行分类，为$（x_n）_ {n \ geq0} $的商集在$ \ mathbb {q} _p $中是密度。在本文中，我们找到了足够的条件，可以使$ k $ th级线性复发$（x_n）_ {n \ geq0} $满足$ x__ {n} = a_1x_ {n-1} = a_1x_ {n-1}使用初始值$ x_0 = \ dots = x_ {k-2} = 0，x_ {k-1} = 1 $，其中$ a_1，\ dots，a_k \ in \ in \ mathbb {z} $和$ a_k = 1 $。我们证明，给定一个Prime $ P $，存在无限的许多复发序列$ k \ geq 2 $，因此它们的商集在$ \ mathbb {q} _p $中不密集。我们还研究了在某些算术和几何发展中具有系数的线性复发序列的商集。
Let $(x_n)_{n\geq0}$ be a linear recurrence of order $k\geq2$ satisfying $$x_n=a_1x_{n-1}+a_2x_{n-2}+\dots+a_kx_{n-k}$$ for all integers $n\geq k$, where $a_1,\dots,a_k,x_0,\dots, x_{k-1}\in \mathbb{Z},$ with $a_k\neq0$. In [`The quotient set of $k$-generalised Fibonacci numbers is dense in $\mathbb{Q}_p$', \emph{Bull. Aust. Math. Soc.} \textbf{96} (2017), 24-29], Sanna posed an open question to classify primes $p$ for which the quotient set of $(x_n)_{n\geq0}$ is dense in $\mathbb{Q}_p$. In this article, we find a sufficient condition for denseness of the quotient set of the $k$th-order linear recurrence $(x_n)_{n\geq0}$ satisfying $ x_{n}=a_1x_{n-1}+a_2x_{n-2}+\dots+a_kx_{n-k}$ for all integers $n\geq k$ with initial values $x_0=\dots=x_{k-2}=0,x_{k-1}=1$, where $a_1,\dots,a_k\in \mathbb{Z}$ and $a_k=1$. We show that given a prime $p$, there exist infinitely many recurrence sequences of order $k\geq 2$ so that their quotient sets are not dense in $\mathbb{Q}_p$. We also study the quotient sets of linear recurrence sequences with coefficients in some arithmetic and geometric progressions.
