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Best Approximations

Published on Saturday, 3rd May 2008, 05:00 am; Solved by 1372;
Difficulty rating: 75%

Problem 192

Let $x$ be a real number.
A best approximation to $x$ for the denominator bound $d$ is a rational number $\frac r s $ in reduced form, with $s \le d$, such that any rational number which is closer to $x$ than $\frac r s$ has a denominator larger than $d$:

$|\frac p q -x | < |\frac r s -x| \Rightarrow q > d$

For example, the best approximation to $\sqrt {13}$ for the denominator bound 20 is $\frac {18} 5$ and the best approximation to $\sqrt {13}$ for the denominator bound 30 is $\frac {101}{28}$.

Find the sum of all denominators of the best approximations to $\sqrt n$ for the denominator bound $10^{12}$, where $n$ is not a perfect square and $ 1 < n \le 100000$.