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Let $N$ be a fixed positive integer that is not a square and $m$ be any nonzero integer. Let $x$ and $y$ be positive integers that solve $$x^2 - N y^2 = m^2$$ with $x + y$ minimal (in light of the comment below, please take $m$ to be 1).

What is known about upper bounds for $x$ and $y$?

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There is a positive constant $c_1$ such that there are values of $N$ with $\log(x+y\sqrt N)>c_1\log\sqrt N$. See, e.g., Yamamoto, Real quadratic number fields with large fundamental units, or this expository essay.

The Yamamoto paper is also available online.

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  • $\begingroup$ So this is a form of lower bound. Do the papers talk about upper bounds? Gerhard "Upper Bounds Useful For Planning" Paseman, 2016.12.10. $\endgroup$ – Gerhard Paseman Dec 11 '16 at 2:37
  • $\begingroup$ Oops. Back to the drawing board. $\endgroup$ – Gerry Myerson Dec 11 '16 at 2:41
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    $\begingroup$ Yamamoto cites the result, $h\log(x+y\sqrt N)<\sqrt N(\log\sqrt N+1)$ where $h$ is the class number, from which it follows that $\log(x+y\sqrt N)<\sqrt N(\log\sqrt N+1)$. The citation is L K Hua, On the least solution of Pell's equation, Bull Amer Math Soc 48 (1942) 731-735. The Hua paper is available at projecteuclid.org/euclid.bams/1183504769 $\endgroup$ – Gerry Myerson Dec 11 '16 at 2:45
  • $\begingroup$ There's a nice paper by Lenstra, uni-oldenburg.de/fileadmin/user_upload/mathe/personen/… $\endgroup$ – Gerry Myerson Dec 11 '16 at 2:54

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