Solving Linear Diophantine Equations

On Solving Linear Diophantine Systems Using Generalized Rosser's Algorithm

Complete Solving of Linear

In this paper, we present an algorithm for solving directly linear Diophantine systems of both equations and inequations. Here directly means without adding slack variables for encoding inequalities as equalities. This algorithm is an extension of the algorithm due to Con-tejean and Devie 9] for solving linear Diophantine systems of equations, which is itself a generalization of the algorithm o...

On Solving Linear Diophantine Systems Using Generalized Rosser’s Algorithm

A difficulty in solving linear Diophantine systems is the rapid growth of intermediate results. Rosser’s algorithm for solving a single linear Diophatine equation is an efficient algorithm that effectively controls the growth of intermediate results. Here, we propose an approach to generalize Rosser’s algorithm and present two algorithms for solving systems of linear Diophantine equations. Then...

Diophantine Equations Related with Linear Binary Recurrences

In this paper we find all solutions of four kinds of the Diophantine equations begin{equation*} ~x^{2}pm V_{t}xy-y^{2}pm x=0text{ and}~x^{2}pm V_{t}xy-y^{2}pm y=0, end{equation*}% for an odd number $t$, and, begin{equation*} ~x^{2}pm V_{t}xy+y^{2}-x=0text{ and}text{ }x^{2}pm V_{t}xy+y^{2}-y=0, end{equation*}% for an even number $t$, where $V_{n}$ is a generalized Lucas number. This pape...

An Efficient Incremental Algorithm for Solving Systems of Linear Diophantine Equations

In this paper, we describe an algorithm for solving systems of linear Diophantine equations based on a generalization of an algorithm for solving one equation due to Fortenbacher 3]. It can solve a system as a whole, or be used incrementally when the system is a sequential accumulation of several subsystems. The proof of termination of the algorithm is diicult, whereas the proofs of completenes...

Solving Linear Diophantine Constraints Incrementally

In this paper, we show how to handle linear Diophantine constraints incre-mentally by using several variations of the algorithm by Contejean and Devie (hereafter called ABCD) for solving linear Diophantine systems 4, 5]. The basic algorithm is based on a certain enumeration of the potential solutions of a system, and termination is ensured by an adequate restriction on the search. This algorith...