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id	title	challengeType	forumTopicId	dashedName
5900f43c1000cf542c50ff4e	Problem 207: Integer partition equations	5	301848	problem-207-integer-partition-equations

--description--

For some positive integers k, there exists an integer partition of the form 4^t = 2^t + k,

where 4^t, 2^t, and k are all positive integers and t is a real number.

The first two such partitions are 4^1 = 2^1 + 2 and 4^{1.584\\,962\\,5\ldots} = 2^{1.584\\,962\\,5\ldots} + 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) = \frac{1}{2}.

In the following table are listed some values of P(m)

\begin{align}
  & P(5) = \frac{1}{1}    \\\\
  & P(10) = \frac{1}{2}   \\\\
  & P(15) = \frac{2}{3}   \\\\
  & P(20) = \frac{1}{2}   \\\\
  & P(25) = \frac{1}{2}   \\\\
  & P(30) = \frac{2}{5}   \\\\
  & \ldots                \\\\
  & P(180) = \frac{1}{4}  \\\\
  & P(185) = \frac{3}{13}
\end{align}$$

Find the smallest $m$ for which $P(m) &lt; \frac{1}{12\\,345}$

# --hints--

`integerPartitionEquations()` should return `44043947822`.

```js
assert.strictEqual(integerPartitionEquations(), 44043947822);
```

# --seed--

## --seed-contents--

```js
function integerPartitionEquations() {

  return true;
}

integerPartitionEquations();
```

# --solutions--

```js
// solution required
```

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--description--

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