A method for finding numerical solutions to Diophantine equations using Spiral Optimization Algorithm with Clustering (SOAC)

Abstract

Diophantine equations are equations containing two or more unknowns, such that only the integer solutions are required. To find solutions of these equations numerically, we can be performed by solving an optimization problem using a metaheuristic method. In this paper, the Spiral Optimization Algorithm with Clustering (SOAC) method is proposed to find solutions to Diophantine equations in the form of polynomial, exponential, and also linear and nonlinear systems of equations. In the implementation of the method on solving some existing benchmark problems, the goal of simulation is to find all solutions only in a single run and in a short period of time. Appropriate values of required parameters are selected during the simulation. Results shows satisfactory in solving four problems in polynomial equations, four problems in exponential equations, and three problems in systems of linear and nonlinear equations. In most of cases, the results yield the same with the analytical or numerical solutions in the reference papers, and in some cases the results give more solutions.