Computational resolution of Diophantine equations by means of alpha‐dense curves

Abstract

Purpose

The purpose of this paper is to use α‐dense curves for solving some Diophantine equations, such as Pythagorean triples, Linear Diophantine equations, the Pell Fermat equation, the Mordell equation for positive values.

Design/methodology/approach

The paper's aim is to present the applications in Number Theory of a new method based on α‐dense curves first developed at the beginning of the 1980s by Yves Cherruault and Arthur Guillez. The α‐dense curves generalize the space filling curves (Peanocurves,…) and fractal curves. This technique can be used for solving all problems of operational research in a simple way. The main idea consists in expressing n variables by means of a single one.

Findings

Apply the method to Number Theory. One of the most important applications is related to global optimization. Multivariable optimization problems coming from operational research or from industry can be quickly and easily solved.

Originality/value

The paper presents a new method based on α‐dense curves for solving Diophantine equations.

Citation

Claudine Bitye Mvondo, E., Cherruault, Y. and Mazza, J. (2012), "Computational resolution of Diophantine equations by means of alpha‐dense curves", Kybernetes, Vol. 41 No. 1/2, pp. 51-67. https://doi.org/10.1108/03684921211213106