标题： 平面上的格点 ax+by+cz=d 和不定方程组 ax+by+cz=d ex+fy+gz=h 显示英文标题

标题： Lattice points on the plane ax+by+cz=d and the diophantine system ax+by+cz=d ex+fy+gz=h

摘要： 本工作涉及的课题是线性三元不定方程ax+by+cz=d (1)，以及不定方程组ax+by+cz=d (2) ex+fy+gz=h，其中系数a,b,c,d,e,f,g,h均为整数。 入门数论书籍通常仅包含如何求解方程(1)的简要说明。 对于方程组(2)，提供的材料更少或几乎没有。 本工作的目的是填补这一空白。 在一些初步的介绍性材料之后，其中包括二元线性不定方程ax+by=c的通解（我们在论文中 later 使用的内容）；我们对方程(1)进行了完整且详细的分析。 我们确定了系数a,b,c,d必须满足的精确条件，以便存在整数解。 然后我们推导出解集的两参数参数描述。 如果方程(1)的解集不为空，则它由所有满足(1)的整数三元组(x,y,z)组成。 从几何上看，这被解释为位于方程(1)所表示平面上的所有格点集合。 同样地，我们通过确定系数必须满足的精确条件来对方程组(2)进行全面分析，以使存在整数解。 我们提供了七个带有详细解答的例子。 显示英文摘要

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.