--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 $P(m) < \frac{1}{12\,345}$

--hints--

integerPartitionEquations() should return 44043947822.

assert.strictEqual(integerPartitionEquations(), 44043947822);

--seed--

--seed-contents--

function integerPartitionEquations() {

  return true;
}

integerPartitionEquations();

--solutions--

// solution required