题目198：歧义数字

欧拉计划

Ambiguous Numbers

A best approximation to a real number x for the denominator bound d is a rational number r/s (in reduced form) with s ≤ d, so that any rational number p/q which is closer to x than r/s has q > d.

Usually the best approximation to a real number is uniquely determined for all denominator bounds. However, there are some exceptions, e.g. 9/40 has the two best approximations 1/4 and 1/5 for the denominator bound 6. We shall call a real number x ambiguous, if there is at least one denominator bound for which x possesses two best approximations. Clearly, an ambiguous number is necessarily rational.

How many ambiguous numbers x = p/q, 0 < x < 1/100, are there whose denominator q does not exceed 10⁸?

题目：

在分母界限为 d 时，定义实数 x 的最佳逼近分数形式为 r/s (已约分)，并且 s ≤ d，那么所有比 r/s 更接近 x 的分数都一定满足 q > d。

一般来说，一个实数的最佳逼近对于所有分母界限都唯一存在。然而，还是有例外，比如 9/40 在界限为 6 时，就有 1/4 和 1/5 这 2 个最佳逼近。如果一个数字有至少一个分母界限，使得它有两个最佳逼近的话，我们就把它叫做歧义数字。很明显，一个歧义数字一定是有理数。

请问，对于条件分母 q 不超过 108， 0 < x < 1/100，存在多少个歧义数字 x = p/q？