Numerical solution of certain exponential and non-linear Diophantine systems of equations by using a discrete particle swarm optimization algorithm

This article proposes the numerical solution of some non-linear systems of Diophantine equations including some exponential cases, by its transformation into an optimization problem. For the first example, the implemented algorithm is able to find all the non-negative integer roots in a given region of the domain of the objective function when compared to the analytic solution. Systems of different sizes are solved, as an example. For these, their answers were found, being them of excellent quality in terms of accuracy and precision, even though there is no way of being certain that they are the only ones following Matiyasevich theorem. However, and for systems of modest size, the computation time increased considerably when the search space was increased.