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Lattice points on the plane ax+by+cz=d and the diophantine system ax+by+cz=d ex+fy+gz=h | | Konstantine Zelator
; | | Date: |
12 May 2008 | | 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. | | Source: | arXiv, 0805.1702 | | Services: | Forum | Review | PDF | Favorites | |
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