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arXiv 1708.03563math.NT
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关于Lucas序列的可积性

On the discriminator of Lucas sequences

Bernadette Faye, Florian Luca, Pieter Moree

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21 pages
发表信息：
Ann. Math. Québec 43 (2019), 51--71
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摘要：

我们考虑由$U_{n+2}(k)=(4k+2)U_{n+1}(k) -U_n(k),$唯一确定的Lucas序列族，初始值为$U_0(k)=0$和$U_1(k)=1$，$k\ge 1$为任意整数.对于任何整数$n\ge 1$，$U_n(k)$的函数$\mathcal{D}_k(n)$被定义为最小整数$m$，使得$U_0(k),U_1(k),\ldots,U_{n-1}(k)$以$m$为模成对不全等。Shallit对$\mathcal{D}_k(n)$的数值计算表明，它有一个相对简单的特征。在本文中，我们将证明，这确实是这样的情况下，显示，对于每个$k\ge 1$有一个常数$n_k$，使${\mathcal D}_{k}(n)$有一个简单的表征，每个$n\ge n_k$。$k=1$案例与$k>1$案例有着根本的不同。

英文摘要：

We consider the family of Lucas sequences uniquely determined by $U_{n+2}(k)=(4k+2)U_{n+1}(k) -U_n(k),$ with initial values $U_0(k)=0$ and $U_1(k)=1$ and $k\ge 1$ an arbitrary integer. For any integer $n\ge 1$ the discriminator function $\mathcal{D}_k(n)$ of $U_n(k)$ is defined as the smallest integer $m$ such that $U_0(k),U_1(k),\ldots,U_{n-1}(k)$ are pairwise incongruent modulo $m$. Numerical work of Shallit on $\mathcal{D}_k(n)$ suggests that it has a relatively simple characterization. In this paper we will prove that this is indeed the case by showing that for every $k\ge 1$ there is a constant $n_k$ such that ${\mathcal D}_{k}(n)$ has a simple characterization for every $n\ge n_k$. The case $k=1$ turns out to be fundamentally different from the case $k>1$.

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