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---
id: 5900f43c1000cf542c50ff4e
title: 'Problem 207: Integer partition equations'
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challengeType: 5
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forumTopicId: 301848
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dashedName: problem-207-integer-partition-equations
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---
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# --description--
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For some positive integers $k$, there exists an integer partition of the form $4^t = 2^t + k$,
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where $4^t$, $2^t$, and $k$ are all positive integers and $t$ is a real number.
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The first two such partitions are $4^1 = 2^1 + 2$ and $4^{1.584\\,962\\,5\ldots} = 2^{1.584\\,962\\,5\ldots} + 6$.
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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 $k ≤ m$.
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Thus $P(6) = \frac{1}{2}$.
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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 $P(m) < \frac{1}{12\\,345}$
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# --hints--
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`integerPartitionEquations()` should return `44043947822` .
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```js
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assert.strictEqual(integerPartitionEquations(), 44043947822);
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```
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# --seed--
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## --seed-contents--
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```js
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function integerPartitionEquations() {
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return true;
}
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integerPartitionEquations();
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```
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# --solutions--
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```js
// solution required
```