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.

  • $\begingroup$ 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. $\endgroup$ – Will Jagy Oct 21 '14 at 18:19
  • $\begingroup$ 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 books.google.com/… .) 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. $\endgroup$ – David E Speyer Oct 21 '14 at 18:39
  • $\begingroup$ 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. $\endgroup$ – David E Speyer Oct 21 '14 at 18:41
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    $\begingroup$ @David, Stillwell attributes a pigeonhole argument for Pell to Dirichlet; the whole thing is available, you can judge how it compares to your idea: books.google.com/… $\endgroup$ – Will Jagy Oct 21 '14 at 19:20
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    $\begingroup$ @WillJagy That's pretty much identical to what I did, thanks! $\endgroup$ – David E Speyer Oct 21 '14 at 19:25

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.


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