An Algorithmic Implementation of Runge’s Method for Cubic Diophantine Equations

Abstract

In modern computer algebra systems (such as Mathematica, Maple, SageMath etc.) algorithms for solving in integers are implemented only for a small number of types of diophantine equations. Usually it means that: 1) linear equations and their systems (with any number of unknowns); 2) quadratic equations in two unknowns; 3) cubic Thue equations in two unknowns. At the same time, there is rather wide class of diophantine equations in two unknowns, for which exists effective solving method (that gives explicit estimates for possible solutions), socalled Runge’s method [10]. Exposition of the standard version Runge’s method can be found in well known books [4] and [9]). However, practical realization of Runge’s method is absent in most computer algebra systems, with the exception of very particular cases (see for example [8]). Too large estimates for the solutions are one of the objective reasons for this. Although they are of polynomial type (see [3, 7, 11]), due to the large exponents occurs useless for computer implementation. The original version of Runge’s method leads to such estimates. It uses the Puiseux expansions at x → ∞ of (the branches of) algebraic function y = Ψ(x), determined by the given diophantine equation