Best Approximations

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:

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.

Written by gamwe6 who lives and works in San Francisco building useful things. You should follow him on Twitter