平面ax + by + cz = d上的格点与丢番图系统ax + by + cz = dex + fy + gz = h
Lattice points on the plane ax+by+cz=d and the diophantine system ax+by+cz=d ex+fy+gz=h
摘要：
本文研究的对象是三元线性丢番图方程ax + by + cz = d（1）和系数a，b，c，d，e，f，g，h为整数的丢番图系统ax + by + cz = d（2）ex + fy + gz = h。介绍性的数论书籍通常只包含如何求解方程（1）的简要概述。在系统（2）上甚至提供更少的材料或不提供材料。这项工作的目的是填补这一空白。经过一些初步的介绍性材料，其中包括两个变量的线性丢番图方程ax + by = c的一般解决方案（材料，我们将在后面的文件），我们提出了一个完整的和详细的分析方程（1）。我们确定的精确条件，系数a，b，c，d必须满足为了整数解的存在。然后，我们推导出一个双参数，参数描述的解决方案。（1）的解集，如果不为空，则由满足（1）的所有整数三元组（x，y，z）组成。在几何学上，这被解释为3-D空间中位于等式（1）的平面上的所有格点的集合。同样，我们通过确定系数必须满足的精确条件来对系统（2）进行详尽的分析，以便存在整数解。我们提供了七个例子，并提供了详细的解决方案。
英文摘要：
The subject matter of this work are the linear, three variable diophantine equation ax+by+cz=d (1), and the diophantine system ax+by+cz=d (2) ex+fy+gz=h with the coefficients a,b,c,d,e,f,g,h being integers. Introductory number theory books, typically contain only a brief outline of how to solve equation (1). Even less or no material is offered on the system (2). The purpose of this work is to fill this gap. After some preliminary, introductory material, which includes the general solution of the two variable linear diophantine equation ax+by=c(material which we use later in the paper); we present a complete and detailed analysis of equation (1). We determine the precise conditions that the coefficients a,b,c,d must satisfy in order for integer solutions to exist. We then derive a two-parameter, parametric description of The solution set. The solution set of (1), if not empty, consists of all integer triples (x,y,z) that satisfy (1). Geometrically, this is interpreted as the set of all lattice points in 3-D space which lie on the plane with equation (1). Similarly, we offer an exhaustive analysis of the system (2) by determining the precise conditions that the coeffients must satisfy in order for integer solutions to exist. We offer seven examples with detailed solutions.
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