A Comparative Study of Continued Fraction and AI Methods for Polynomial Pell Equations

Abstract

In this paper, we provide an overview of the comparative analysis of solving the Pell’s equation x2−Dy2 = 1 and the corresponding Pell’s type equation x2 − Dy2 = −1, where D is a given quadratic polynomial that is not a perfect square, by using tools based on generative artificial intelligence. The use of artificial intelligence today has a large and diverse application in society and science. It is important to carefully consider when, to what extent and whether to use artificial intelligence tools in some learning, research or analysis. Therefore, it is very meaningful to test AI tools in its wide range of applications. The use of artificial intelligence in mathematics is not a new form of research, but more modern and technologically improved with the respect to the previously used tools. In this paper, we compared the success of using AI in solving equations x2 − dy2 = 1 and x2 − dy2 = −1. The analysis clearly indicates that both tools still show a high degree of vagueness, a poor strategy and a uniform approach in the selection of methods for solving equations. Since ChatGPT and Gemini are one of the most available artificial intelligence tools a comparison of the solutions provided by these tools is conducted. In addition to the given artificial intelligence solutions, we also present classical mathematical solutions. The number of correct, incorrect and undetermined answers is used as a measurement for the success these tools in solving the mentioned equations.