## Integer partition equations

### Problem 207

Published on Saturday, 6th September 2008, 02:00 pm; Solved by 3269; Difficulty rating: 40%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