---
id: 5900f43c1000cf542c50ff4e
challengeType: 5
title: 'Problem 207: Integer partition equations'
---

## Description
<section id='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
</section>

## Instructions
<section id='instructions'>

</section>

## Tests
<section id='tests'>

```yml
tests:
  - text: <code>euler207()</code> should return 44043947822.
    testString: assert.strictEqual(euler207(), 44043947822, '<code>euler207()</code> should return 44043947822.');

```

</section>

## Challenge Seed
<section id='challengeSeed'>

<div id='js-seed'>

```js
function euler207() {
  // Good luck!
  return true;
}

euler207();
```

</div>



</section>

## Solution
<section id='solution'>

```js
// solution required
```
</section>
