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On the Diophantine Equations
Abstract
An algorithm is devised to determine all solutions of any Diophantine equation of the type described in the title.
... In 1970, Matijasevic [7] proved the non-existence of such an algorithm. Meanwhile, researchers looked for a general algorithm for solving Diophantine equations of special forms [4]. However, in this research, we aim to test the success rate of a special algorithm in solving Diophantine equations of general form. ...
In this paper, experimental results for presenting the success rates in finding integral solutions to the Diophantine equations of two variables using modular arithmetic are shown. Firstly, and for this purpose, the empirical algorithm for obtaining experimental results is presented for Diophantine equations of polynomial-type. Using modular arithmetic, the algorithm transforms the Diophantine equation to a linear recurrence relation. This defines a sequence of integers { s } s=1 and the integral solution is achieved once this sequence terminates. The algorithm could be successful in solving polynomial Diophantine equations provided that a predefined numeral system pattern to the base p (prime number) for one of the two variables exists. Secondly, without any predefined numeral system pattern, the algorithm is adopted for equations with exponential terms. Finally, and for experimental results, four strategies are proposed for the termination of the sequence { s } s. Success rates are compared by testing the strategies over many polynomial-type Diophantine equations. The experimental results are presented in detail and some strategies contribute to high success rates in achieving the integral solutions.