90 lines
5.7 KiB
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90 lines
5.7 KiB
HTML
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<h2>Integer partition equations</h2><div class="info" style="cursor:help;width:200px;margin-bottom:10px;"><h3>Problem 207</h3><span style="width:300px;color:#666;">Published on Saturday, 6th September 2008, 02:00 pm; Solved by 2719</span></div>
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<p>For some positive integers <var>k</var>, there exists an integer partition of the form 4<sup>t</sup> = 2<sup>t</sup> + <var>k</var>,<br />
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where 4<sup>t</sup>, 2<sup>t</sup>, and <var>k</var> are all positive integers and <var>t</var> is a real number.</p>
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<p>The 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.</p>
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<p>Partitions where <var>t</var> is also an integer are called <i>perfect</i>.<br />
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For any <var>m</var> <img src='images/symbol_ge.gif' width='10' height='12' alt='≥' border='0' style='vertical-align:middle;' /> 1 let P(<var>m</var>) be the proportion of such partitions that are perfect with <var>k</var> <img src='images/symbol_le.gif' width='10' height='12' alt='≤' border='0' style='vertical-align:middle;' /> <var>m</var>.<br />
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Thus P(6) = 1/2.</p>
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<p>In the following table are listed some values of P(<var>m</var>)</p>
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<p> P(5) = 1/1<br />
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P(10) = 1/2<br />
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P(15) = 2/3<br />
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P(20) = 1/2<br />
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P(25) = 1/2<br />
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P(30) = 2/5<br />
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...<br />
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P(180) = 1/4<br />
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P(185) = 3/13</p>
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<p>Find the smallest <var>m</var> for which P(<var>m</var>) <img src='images/symbol_lt.gif' width='10' height='10' alt='<' border='0' style='vertical-align:middle;' /> 1/12345</p>
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