On periodic continued fractions, Pell equation, and Fermat's challenge numbers

Article type: Research Article

Authors: Mestechkin, M.

Affiliations: 12773 Seabreeze Farms Dr. # 33, San Diego CA 92130, USA. E-mail: mestechkinmm@gmail.com

Abstract: A method for direct construction of continued fraction elements, presenting a square root from a natural number, is proposed. This technique resembles the usual division process. Derivation of the computing procedures is based on the Euclidean algorithm, Fermat's and Euler's theorems on a sum of two squares, and formulation of the basic equation for the continued fraction elements as Diophantine equation. The solution of Pell equation is presented explicitly through the same functions, which describe the Euclidean algorithm and solutions to linear Diophantine equations. It is supposed that similar procedures could be used by Fermat when he challenged to solve the quadratic Diophantine equation for special difficult cases found by him.

Keywords: Continued fraction, Pell Equation, Euclidean algorithm, Diophantine equation

DOI: 10.3233/JCM-2010-0261

Journal: Journal of Computational Methods in Sciences and Engineering, vol. 10, no. 1-2, pp. 49-66, 2010