## Best Approximations

### Problem 192

Published on Saturday, 3rd May 2008, 05:00 am; Solved by 890

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 1012, where n is not a perfect square and 1 n 100000.