论文标题
晶格平等四边形I-平行四边形
Lattice Equable Quadrilaterals I -- Parallelograms
论文作者
论文摘要
本文研究其顶点位于整数晶格上的公平平行四边形。使用Rosenberger的定理在广义Markov方程式上，我们表明G.C.D.在此类平行四边形的侧面长度中，只能是3、4或5，在每种情况下，平行四边形的集合自然形成无限树，其顶点具有4度为4，bar the root the root。然后，本文关注我们所谓的毕达哥拉斯等于平等的平行四边形。这些是晶格平等的平行四边形，其在围绕矩形中的补充由两个毕达哥拉斯三角形组成。我们证明，对于这些平行四边形，最短的一侧只能是3、4、5、6或10，并且有五个无限的平行四边形族，由对相应的类似佩尔样方程的溶液给出。
This paper studies equable parallelograms whose vertices lie on the integer lattice. Using Rosenberger's Theorem on generalised Markov equations, we show that the g.c.d. of the side lengths of such parallelograms can only be 3, 4 or 5, and in each of these cases the set of parallelograms naturally forms an infinite tree all of whose vertices have degree 4, bar the root. The paper then focuses on what we call Pythagorean equable parallelograms. These are lattice equable parallelograms whose complement in a circumscribing rectangle consists of two Pythagorean triangles. We prove that for these parallelograms the shortest side can only be 3, 4, 5, 6 or 10, and there are five infinite families of such parallelograms, given by solutions to corresponding Pell-like equations.
