Problem 207: Integer partition equations

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

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

The first two such partitions are 41 = 21 + 2 and 41.5849625... = 21.5849625... + 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