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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$\begingroup$ Peek at imomath.com/index.php?options=615, for instance. $\endgroup$– vonbrandCommented Jul 23, 2015 at 1:16
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5$\begingroup$ If you don't know how it fits into your course, and why you'd want to consider it, better leave it out... $\endgroup$– vonbrandCommented Jul 23, 2015 at 1:18
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2$\begingroup$ 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. CookCommented Dec 6, 2015 at 3:33
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1 Answer
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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 https://kconrad.math.uconn.edu/.)
answered Nov 6, 2015 at 0:39
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2$\begingroup$ See also kconrad.math.uconn.edu/blurbs/ugradnumthy/pelleqn1.pdf and kconrad.math.uconn.edu/blurbs/ugradnumthy/pelleqn2.pdf $\endgroup$– KCdCommented May 25 at 21:47