# Solving Pell's equation using continued fraction

Lee, Yee Shyuan (2007) Solving Pell's equation using continued fraction. Universiti Malaysia Sabah. (Unpublished)

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## Abstract

The main aim of this dissertation is to generalize and to apply results using infinite continued fraction to represent and accurately disclose the solution to a Pell's equation. The Theory of Pell's equation was added to provide a background of this equation. The wide field of continued fraction has been narrowed down and focuses only on infinite continued fraction which deduced the expansion of the square root of a positive irrational integer (âˆšd) for a Pell's equation xÂ² - dyÂ² = 1. Two other methods, namely substitution method and residual method. provide another alternative to the expansion. Other positive solutions of Pell's equation can be generated from the fundamental solution using a new method found. The relationship between two positive irrational square root in the continued fraction table of irrational square root from âˆš2 to âˆš99 (Refer Appendix A) was determined and proved to apply to all irrational square roots when the properties stated were fulfilled. Verification properties to affirm the accuracy of the expansion of infinite continued fraction was discovered and proven so that there is a checklist to follow especially if the expansion is long and difficult. Three Excel worksheets inside one spreadsheet are included to efficiently disclose the quotients, sequences and also to generate other positive solutions of Pell's equation.

Item Type: Academic Exercise Continued Fraction, Pell's equation, Indian Mathematicians, substitution method Q Science > QA Mathematics SCHOOL > School of Science and Technology MDM SITI AZIZAH IDRIS 15 Jul 2014 00:44 17 Oct 2017 04:38 http://eprints.ums.edu.my/id/eprint/7333