Integer partition equations

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 $4^{1.5849625...} = 2^{1.5849625...} + 6$.

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 $P(m) < 1/12345$