Abstract
This paper uses a continued fraction to explain various approaches to solving Diophantine equations. It first examines the fundamental characteristics of continued fractions, such as convergent and approximations to real numbers. Using continued fractions, we can solve the Pell's equation. Certain theorems have also been discussed for how to expand quadratic irrational integers into periodic continued fractions. Finally, the relationship between convergent and best approximations and use of continuous fraction in calendar construction has-been investigated. The analytical theory of continued fractions is a significant generalization of continued fractions and represents a large field for current and future research.
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