具有整数坐标的等边三角形的参数化
A parametrization of equilateral triangles having integer coordinates
摘要：
研究了三维空间中给定边长且坐标为整数的等边三角形的存在性。我们证明了这样一个三角形存在的充要条件是它们的边长对于某些整数$m,n$是$\sqrt{2(m^2-mn+n^2)}$的形式。我们还显示了一个类似的特征，在$\Z^3$的正四面体的侧面：这样的四面体存在，当且仅当边的形式为$k\sqrt{2}$，对于一些$k\in\N$。研究了包含在给定平面内的$\Z^3$中所有等边三角形的分类问题，并给出了初步的分析。一个更一般的参数化证明下一个特殊的假设。最后提出了一些相关问题。
英文摘要：
We study the existence of equilateral triangles of given side lengths and with integer coordinates in dimension three. We show that such a triangle exists if and only if their side lengths are of the form $\sqrt{2(m^2-mn+n^2)}$ for some integers $m,n$. We also show a similar characterization for the sides of a regular tetrahedron in $\Z^3$: such a tetrahedron exists if and only if the sides are of the form $k\sqrt{2}$, for some $k\in\N$. The classification of all the equilateral triangles in $\Z^3$ contained in a given plane is studied and the beginning analysis is presented. A more general parametrization is proven under a special assumption. Some related questions are stated in the end.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
