How to solve equations using continued fractions?

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SUMMARY

Continued fractions can be effectively utilized to solve polynomial equations, particularly for obtaining roots of low-degree polynomials. They are known to generate rapidly converging sequences that approximate the roots of equations, including applications in differential equations. The discussion highlights the relevance of Pell's theorem and references a specific application in exponential Diophantine equations. However, for polynomial equations of degree five and higher, numerical methods are typically preferred over continued fractions.

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  • Understanding of continued fractions and their properties
  • Familiarity with polynomial equations and their roots
  • Knowledge of Pell's theorem and its applications
  • Basic concepts of numerical methods for solving equations
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  • Research the application of continued fractions in solving polynomial equations
  • Explore the role of Pell's theorem in number theory
  • Learn about numerical methods for solving higher-degree polynomial equations
  • Investigate the use of continued fractions in differential equations
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Arian.D
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Is it possible to solve any equation if we use continued fractions? I've heard that polynomial equations could be solved using continued fractions, and I used to obtain one of the several roots of a polynomial equation of low degrees using continued fractions in high school, but I read somewhere that continued fractions could be employed to find rapid converging sequences to the roots of an equation and I guess I even read somewhere that they could be used to solve differential equations!

Could someone inform me about the use of continued fractions to solve equations please? Thanks in advance.
 
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I've found some applications of continued fractions ( http://www.rnta.eu/SecondRNTA/Waldschmidt-Sanna.pdf ) and in section 5 exponential Diophantine equations. Also Pell's theorem plays a role, but I haven't seen applications on general roots. I guess there are better algorithms, and from degree 5 onwards we only have numerical methods.
 

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