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arXiv:2503.01911 (math)
[Submitted on 1 Mar 2025 ]

Title: Planar lattices and equilateral odd-gons

Title: 平面格点和等边奇数边形

Authors:Akira Iino, Masashi Sakiyama
Abstract: For a planar integral lattice $L$, let $\nu(L)$ denote the square-free part of the integer $D(L)^2$, where $D(L)$ stands for the area of a fundamental parallelogram of $L$. For each odd integer $n$ with $3 \leq n<29$, a planar lattice $L$ contains an equilateral $n$-gon if and only if $L$ is similar to an integral lattice $L'$ such that $\nu(L')\equiv 3 \pmod 4$ and the largest prime factor $p$ of $\nu(L')$ satisfies $p \leq n$. Moreover, such $L$ contains a convex equilateral $n$-gon, which answers a problem posed by Maehara.
Abstract: 对于一个平面整数格点$L$,令$\nu(L)$表示整数$D(L)^2$的平方自由部分,其中$D(L)$表示$L$的一个基本平行四边形的面积。 对于每个奇整数$n$且$3 \leq n<29$,一个平面格点$L$包含一个等边$n$-边形当且仅当$L$与一个整格点$L'$相似,使得$\nu(L')\equiv 3 \pmod 4$且$\nu(L')$的最大素因数$p$满足$p \leq n$。 此外,这样的$L$包含一个凸等边$n$-边形,这回答了 Maehara 提出的一个问题。
Comments: 7 pages, to appear in Yokohama Mathematical Journal
Subjects: Combinatorics (math.CO) ; Metric Geometry (math.MG); Number Theory (math.NT)
MSC classes: 52C05, 11H06, 51M04
Cite as: arXiv:2503.01911 [math.CO]
  (or arXiv:2503.01911v1 [math.CO] for this version)
  https://doi.org/10.48550/arXiv.2503.01911
arXiv-issued DOI via DataCite
Journal reference: Yokohama Math. J. 70 (2024) 235-242
Related DOI: https://doi.org/10.18880/0002001755
DOI(s) linking to related resources

Submission history

From: Masashi Sakiyama [view email]
[v1] Sat, 1 Mar 2025 04:59:41 UTC (7 KB)
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