An Improved Algorithm for Diophantine Equations in One Variable

"Syntactic" AC-Unification

The rules for unification in a simple syntactic theory, using Kirchner’s mutation [15, 16] do not terminate in the case of associativecommutative theories. We show that in the case of a linear equation, these rules terminate, yielding a complete set of solved forms, each variable introduced by the unifiers corresponding to a (trivial) minimal solution of the (trivial) Diophantine equation where...

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...

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...

On Solving Linear Diophantine Systems Using Generalized Rosser's Algorithm

Solving Systems of Linear Diophantine Equations: An Algebraic Approach

We describe through an algebraic and geometrical study, a new method for solving systems of linear diophantine equations. This approach yields an algorithm which is intrinsically parallel. In addition to the algorithm, we give a geometrical interpretation of the satissability of an homogeneous system, as well as upper bounds on height and length of all minimal solutions of such a system. We als...

on ”NUMBER THEORY AND MATHEMATICAL PHYSICS” On recent Diophantine results

Diophantus of Alexandria was a greek mathematician, around 200 AD, who studied mathematical problems, mostly geometrical ones, which he reduced to equations in rational integers or rational numbers. He was interested in producing at least one solution. Such equations are now called Diophantine equations. An example is y − x = 1, a solution of which is (x = 2, y = 3). More generally, a Diophanti...