# Upper bounds for solutions to a Pell-like equation

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 $y$?

• They are at most $m$ times the fundamental solution to $x^2-Ny^2=1$, and in general we cannot do better. – Wojowu Dec 5 '16 at 19:18
• Perhaps this is a duplicate of mathoverflow.net/questions/252879/… – Gerry Myerson Dec 12 '16 at 0:22
• – Gerry Myerson Dec 12 '16 at 0:27

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.
• 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 – Gerry Myerson Dec 11 '16 at 2:45