COMPLETE CLASSIFICATION OF QUADRATIC IRRATIONALS WITH PERIOD TWO

This article presents a comprehensive investigation into the classification of quadratic irrationals with period two in their continued fraction representations. Building upon foundational results in Number Theory, particularly in the context of continued fractions and Pell's equation, the study reveals intricate relationships between quadratic irrationals and their periodic structures. The main object of study is √N and properties of its continued fractions. While it is well-known that continued fractions of √N is periodic with periodic part being palindrome, the distribution of the lengths of the periodic parts are far from being complete. Our main goal will be to focus on the period two case and provide a complete characterization. The research's proved theorems clarify the conditions under which the period length is exactly two and give an insight into the underlying algebraic features. Additionally, it delves deeper by offering numerical analysis and illustrations demonstrating the distribution of period lengths among quadratic irrationals. This research opens up new paths for future studies on quadratic irrationals and how they're shown as continued fractions.