47 lines
1.0 KiB
Markdown
47 lines
1.0 KiB
Markdown
---
|
|
id: 5900f42c1000cf542c50ff3f
|
|
title: 'Problem 192: Best Approximations'
|
|
challengeType: 5
|
|
forumTopicId: 301830
|
|
dashedName: problem-192-best-approximations
|
|
---
|
|
|
|
# --description--
|
|
|
|
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.
|
|
|
|
# --hints--
|
|
|
|
`euler192()` should return 57060635927998344.
|
|
|
|
```js
|
|
assert.strictEqual(euler192(), 57060635927998344);
|
|
```
|
|
|
|
# --seed--
|
|
|
|
## --seed-contents--
|
|
|
|
```js
|
|
function euler192() {
|
|
|
|
return true;
|
|
}
|
|
|
|
euler192();
|
|
```
|
|
|
|
# --solutions--
|
|
|
|
```js
|
|
// solution required
|
|
```
|