yaworsw/euler-manager

---
:id: 207
:name: Integer partition equations
:url: https://projecteuler.net/problem=207
:content: "For some positive integers <var>k</var>, there exists an integer partition
  of the form&nbsp;&nbsp; 4<sup>t</sup> = 2<sup>t</sup> + <var>k</var>,  \nwhere 4<sup>t</sup>,
  2<sup>t</sup>, and <var>k</var> are all positive integers and <var>t</var> is a
  real number.\n\nThe first two such partitions are 4<sup>1</sup> = 2<sup>1</sup>
  + 2 and 4<sup>1.5849625...</sup> = 2<sup>1.5849625...</sup> + 6.\n\nPartitions where
  <var>t</var> is also an integer are called _perfect_.  \n For any <var>m</var> ≥
  1 let P(<var>m</var>) be the proportion of such partitions that are perfect with
  <var>k</var> ≤ <var>m</var>.  \nThus P(6) = 1/2.\n\nIn the following table are listed
  some values of P(<var>m</var>)\n\n&nbsp;&nbsp;&nbsp;P(5) = 1/1  \n&nbsp;&nbsp;&nbsp;P(10)
  = 1/2  \n&nbsp;&nbsp;&nbsp;P(15) = 2/3  \n&nbsp;&nbsp;&nbsp;P(20) = 1/2  \n&nbsp;&nbsp;&nbsp;P(25)
  = 1/2  \n&nbsp;&nbsp;&nbsp;P(30) = 2/5  \n&nbsp;&nbsp;&nbsp;...  \n&nbsp;&nbsp;&nbsp;P(180)
  = 1/4  \n&nbsp;&nbsp;&nbsp;P(185) = 3/13\n\nFind the smallest <var>m</var> for which
  P(<var>m</var>) \\< 1/12345\n\n"