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

TL;DR

This paper thoroughly analyzes the solutions of linear Diophantine equations and systems with three variables, providing conditions for solutions and a parametric description, filling gaps in existing number theory literature.

Contribution

It offers a complete, detailed analysis of the three-variable Diophantine equation and system, including solution conditions and geometric interpretation, which were previously underexplored.

Findings

Derived necessary and sufficient conditions for solutions to exist.

Provided a parametric description of the solution set.

Included detailed examples illustrating the solution process.

Abstract

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.