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David Speyer gave a beautiful application of the pigeon hole principle here to show that Pell's equation $$x^2-Dy^2=1$$ has infinitely many integral solutions.

I was wondering if anybody knows the history/origin of this argument...In particular was this the original argument used by Lagrange? Or was it Dirichlet? Or is this argument an original due to Speyer? Thanks!

PS I am not looking for alternative proofs of the solvability of Pell's equation...just comments on the proof given above.

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Chris McDaniel
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  • left a comment at that answer for David, with a link back to here. He did leave a comment earlier today, so perhaps he will notice my comment today. – Will Jagy Oct 21 '14 at 18:19
  • I came up with this formulation about ten years ago after reading a proof of Dirichlet's Unit Theorem and trying to specialize it to real quadratic field. (It might have been Borevich and Shavarevich http://books.google.com/books?id=njgVUjjO-EAC&pg=PA112#v=onepage&q&f=false .) I don't think there is much original here. Most (perhaps all?) proofs of Pell's theorem use something like this technique; what might be original to me is emphasizing how this is pigeonhole over and over. – David E Speyer Oct 21 '14 at 18:39
  • For example, the continued fraction proof uses that there are finitely many reduced quadratic forms of discriminant $D$ to deduce that the continued fraction is periodic, and then compares convergents separated by a cycle to solve the equation. That's basically the same thing, except that I notice you can just talk about solutions to $|p^2-D q^2| \leq 2 \sqrt{D}$ without bringing up the whole theory of continued fractions. – David E Speyer Oct 21 '14 at 18:41
  • I'd certainly be glad to see a more authoritative answer by someone who knows history better than I do. – David E Speyer Oct 21 '14 at 18:41
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    @David, Stillwell attributes a pigeonhole argument for Pell to Dirichlet; the whole thing is available, you can judge how it compares to your idea: http://books.google.com/books?id=LiAlZO2ntKAC&pg=PA84&lpg=PA84&dq=pell+pigeonhole&source=bl&ots=Pq-8X1zvg4&sig=9V-EvNdGVH3LHf6dEj1d3IoJ9O8&hl=en&sa=X&ei=7LBGVO2VGM6UyASdvYD4CQ&ved=0CDYQ6AEwAw#v=onepage&q=pell%20pigeonhole&f=false – Will Jagy Oct 21 '14 at 19:20
  • note that I just put Pell Pigeonhole into google – Will Jagy Oct 21 '14 at 19:24
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    @WillJagy That's pretty much identical to what I did, thanks! – David E Speyer Oct 21 '14 at 19:25
  • To get a truly authoritative answer, we need someone to track down Dirichlet's paper and see that this actually is what he did. – David E Speyer Oct 21 '14 at 19:28
  • @David, thanks for the comments, and thanks for posting that beautiful application of the pigeon hole principle! – Chris McDaniel Oct 21 '14 at 20:29
  • @Will, thanks for the comments, and thanks for that Stillwell reference...that's just what I was looking for. – Chris McDaniel Oct 21 '14 at 20:31

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Dirichlet's proof using the pigeonhole principle is a simplification of Lagrange's proof, but actually the pigeonhole principle already appears in Lagrange. Dirichlet's simplification was replacing an argument involving continued fractions by invoking the pigeonhole principle a second time.