# Problem 207: Integer partition equations

![graphic](img207.gif)

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