49 lines
1.0 KiB
Markdown
49 lines
1.0 KiB
Markdown
---
|
|
id: 5900f43c1000cf542c50ff4e
|
|
title: 'Problem 207: Integer partition equations'
|
|
challengeType: 5
|
|
forumTopicId: 301848
|
|
dashedName: problem-207-integer-partition-equations
|
|
---
|
|
|
|
# --description--
|
|
|
|
For some positive integers k, there exists an integer partition of the form 4t = 2t + k,
|
|
|
|
where 4t, 2t, and k are all positive integers and t is a real number.
|
|
|
|
The first two such partitions are 41 = 21 + 2 and 41.5849625... = 21.5849625... + 6.
|
|
|
|
Partitions where t is also an integer are called perfect. For any m ≥ 1 let P(m) be the proportion of such partitions that are perfect with k ≤ m. Thus P(6) = 1/2.
|
|
|
|
In the following table are listed some values of P(m) P(5) = 1/1 P(10) = 1/2 P(15) = 2/3 P(20) = 1/2 P(25) = 1/2 P(30) = 2/5 ... P(180) = 1/4 P(185) = 3/13
|
|
|
|
Find the smallest m for which P(m) < 1/12345
|
|
|
|
# --hints--
|
|
|
|
`euler207()` should return 44043947822.
|
|
|
|
```js
|
|
assert.strictEqual(euler207(), 44043947822);
|
|
```
|
|
|
|
# --seed--
|
|
|
|
## --seed-contents--
|
|
|
|
```js
|
|
function euler207() {
|
|
|
|
return true;
|
|
}
|
|
|
|
euler207();
|
|
```
|
|
|
|
# --solutions--
|
|
|
|
```js
|
|
// solution required
|
|
```
|