AI 中文总结
本文证明了 Kimberling 关于数组 $\lfloor k\varphi^n\rfloor$ 的三个猜想：前两个由 Skolem-Bang 定理推出，第三个通过确定包含奇下标 Lucas 数的行及 Lucas 移位引理证明，并给出计算证据。
AI 中文摘要
设 $\varphi$ 为黄金比例，并设 $R_n=\{\lfloor k\varphi^n\rfloor: k\ge 1\}$ 为数组 $T(n,k)=\lfloor k\varphi^n\rfloor$（OEIS A128440）的第 $n$ 行。2022年，Kimberling 猜想行 $R_{2n-1}$ 和 $R_{2n}$ 是不相交的，并且在这两行合并且每个条目被其秩替换后，它们成为下和上 Wythoff 序列。他还猜想（OEIS A358359）如果 $a(N)$ 是包含 $N$ 的行数，那么每个正整数在 $a$ 的值中无限次出现。我们证明前两个猜想可以快速从 Skolem-Bang 定理得出，该定理也给出了两行不相交的精确规则：$R_i\cap R_j=\emptyset$（$i<j$）当且仅当 $j-i$ 为奇数且整除 $i$。然后我们证明第三个猜想。主要工具是对包含奇下标 Lucas 数的行的显式确定，这推广了 Noppakaew、Kanwarunyu 和 Wanitchatchawan 的一个结果，以及一个“Lucas 移位”引理：如果 $N+1$ 不是形如 $L_{2e}$（$e\ge1$）的数，那么将足够大的偶下标 Lucas 数加到 $N$ 上不会改变包含它的行集合。我们还证明 $a$ 的每个值在一个具有正自然密度的集合上被取到，并报告了高达 $10^8$ 的计算，表明恰好位于 $v\ge 2$ 行中的最小 $N$ 是 Lucas 数 $L_{4v-5}$。
英文摘要
Let $φ$ be the golden ratio and let $R_n=\{\lfloor kφ^n\rfloor : k\ge 1\}$ be the $n$-th row of the array $T(n,k)=\lfloor kφ^n\rfloor$ (OEIS A128440). In 2022 Kimberling conjectured that the rows $R_{2n-1}$ and $R_{2n}$ are disjoint, and that after the two rows are merged and each entry is replaced by its rank, they become the lower and upper Wythoff sequences. He also conjectured (OEIS A358359) that if $a(N)$ is the number of rows containing $N$, then every positive integer occurs infinitely often among the values of $a$. We show that the first two conjectures follow quickly from the Skolem-Bang theorem, which also yields the exact rule for when two rows are disjoint: $R_i\cap R_j=\emptyset$ ($i<j$) if and only if $j-i$ is odd and divides $i$. We then prove the third conjecture. The main tools are an explicit determination of the rows containing an odd-indexed Lucas number, which extends a result of Noppakaew, Kanwarunyu and Wanitchatchawan, and a "Lucas shift" lemma: if $N+1$ is not of the form $L_{2e}$ with $e\ge1$, then adding a sufficiently large even-indexed Lucas number to $N$ does not change the set of rows containing it. We also show that each value of $a$ is taken on a set of positive natural density, and we report computations up to $10^8$ suggesting that the least $N$ lying in exactly $v\ge 2$ rows is the Lucas number $L_{4v-5}$.
Comments7 pages. Also deposited at Zenodo, doi:10.5281/zenodo.23124803
