What is the best way to introduce Pell’s equation on a first elementary number theory course? Are there any practical applications of Pell’s equation? What are the really interesting questions about Pell’s equation? Are there any good resources on Pell’s equation.

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    $\begingroup$ Have you tried a web search? Gerhard "Really, Have You Tried It?" Paseman, 2015.07.15 $\endgroup$ – Gerhard Paseman Jul 15 '15 at 22:14
  • $\begingroup$ Peek at imomath.com/index.php?options=615, for instance. $\endgroup$ – vonbrand Jul 23 '15 at 1:16
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    $\begingroup$ If you don't know how it fits into your course, and why you'd want to consider it, better leave it out... $\endgroup$ – vonbrand Jul 23 '15 at 1:18
  • $\begingroup$ There is a nice chapter in Stillwell's number theory text. It has a bit about the rational approximation mentioned in the answer below. Also, the problem of finding the first solution from which the others can be generated is considered in that chapter. $\endgroup$ – James S. Cook Dec 6 '15 at 3:33

Keith Conrad gave a presentation in 2008 that addresses your question, e.g.:

Pell solutions lead to good rational approximations to $\sqrt{d}$:

          (PDF download presentation from http://www.math.uconn.edu/~kconrad/.)

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