Best Approximations
Problem 192
Published on Saturday, 3rd May 2008, 05:00 am; Solved by 946; Difficulty rating: 75%Let x be a real number.
A best approximation to x for the denominator bound d is a rational number r/s in reduced form, with s ≤ d, such that any rational number which is closer to x than r/s has a denominator larger than d:
|p/q-x| < |r/s-x| ⇒ q > d
For example, the best approximation to √13 for the denominator bound 20 is 18/5 and the best approximation to √13 for the denominator bound 30 is 101/28.
Find the sum of all denominators of the best approximations to √n for the denominator bound 10^{12}, where n is not a perfect square and 1 < n ≤ 100000.