发表机构
Research Affairs, Faculty of Medicine, Chulalongkorn University（朱拉隆功大学医学院科研事务处）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对Soifer猜想，开发新工具证明了当n=4时，无法用n^2+1个单位正方形覆盖边长大于n的正方形，并可能推广至相关问题。
AI 中文摘要
2006年，Alexander Soifer猜想：无法用 $n^2 + O(1)$ 个单位正方形完全覆盖边长大于 $n$ 的正方形。对于较小的 $n \in \{2, 3\}$，2009年，Janusz Januszewski证明了：无法用恰好 $n^2 + 1$ 个单位正方形完全覆盖边长大于 $n$ 的正方形。最近在2023年，Baek和Lee证明了：无法用恰好 $n^2 + 1$ 个边与其平行的单位等边三角形完全覆盖边长大于 $n$ 的等边三角形。在相关问题上也取得了一些进展：用 $k$ 个单位正方形能完全覆盖的最大正方形的边长 $S(k)$ 是多少。然而，原始猜想一直没有改进。在这项工作中，我们开发了新工具，从而证明了该猜想在 $n = 4$ 时成立，并可能应用于相关问题。
英文摘要
In 2006, Alexander Soifer conjectured that one cannot fully cover a square of side length $> n$ with $n^2 + O(1)$ unit squares. For small $n \in \{2, 3\}$, in 2009, Janusz Januszewski proved that it is impossible to fully cover a square of side length $> n$ with exactly $n^2 + 1$ unit squares. Recently in 2023, Baek and Lee proved that it is impossible to fully cover an equilateral triangle of side length $> n$ with exactly $n^2 + 1$ unit equilateral triangles whose sides are parallel to it. There were also some progress on a related problem: what is the largest square with side length $S(k)$ that can be fully covered by $k$ unit squares. However, there have been no improvement on the original conjecture. In this work, new tools have been developed that led to the proof of this conjecture for $n = 4$, with potential applications to related problems.
Comments9 pages, 6 figures, intended for journal publication
