86 lines
7.5 KiB
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86 lines
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<h2>Efficient exponentiation</h2><div class="info" style="cursor:help;width:200px;margin-bottom:10px;"><h3>Problem 122</h3><span style="width:300px;color:#666;">Published on Friday, 2nd June 2006, 06:00 pm; Solved by 3681</span></div>
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<p>The most naive way of computing <i>n</i><sup>15</sup> requires fourteen multiplications:</p>
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<p style='margin-left:100px;'><i>n</i> <img src='images/symbol_times.gif' width='9' height='9' alt='×' border='0' style='vertical-align:middle;' /> <i>n</i> <img src='images/symbol_times.gif' width='9' height='9' alt='×' border='0' style='vertical-align:middle;' /> ... <img src='images/symbol_times.gif' width='9' height='9' alt='×' border='0' style='vertical-align:middle;' /> <i>n</i> = <i>n</i><sup>15</sup></p>
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<p>But using a "binary" method you can compute it in six multiplications:</p>
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<p style='margin-left:100px;'><i>n</i> <img src='images/symbol_times.gif' width='9' height='9' alt='×' border='0' style='vertical-align:middle;' /> <i>n</i> = <i>n</i><sup>2</sup><br />
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<i>n</i><sup>2</sup> <img src='images/symbol_times.gif' width='9' height='9' alt='×' border='0' style='vertical-align:middle;' /> <i>n</i><sup>2</sup> = <i>n</i><sup>4</sup><br />
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<i>n</i><sup>4</sup> <img src='images/symbol_times.gif' width='9' height='9' alt='×' border='0' style='vertical-align:middle;' /> <i>n</i><sup>4</sup> = <i>n</i><sup>8</sup><br />
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<i>n</i><sup>8</sup> <img src='images/symbol_times.gif' width='9' height='9' alt='×' border='0' style='vertical-align:middle;' /> <i>n</i><sup>4</sup> = <i>n</i><sup>12</sup><br />
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<i>n</i><sup>12</sup> <img src='images/symbol_times.gif' width='9' height='9' alt='×' border='0' style='vertical-align:middle;' /> <i>n</i><sup>2</sup> = <i>n</i><sup>14</sup><br />
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<i>n</i><sup>14</sup> <img src='images/symbol_times.gif' width='9' height='9' alt='×' border='0' style='vertical-align:middle;' /> <i>n</i> = <i>n</i><sup>15</sup></p>
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<p>However it is yet possible to compute it in only five multiplications:</p>
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<p style='margin-left:100px;'><i>n</i> <img src='images/symbol_times.gif' width='9' height='9' alt='×' border='0' style='vertical-align:middle;' /> <i>n</i> = <i>n</i><sup>2</sup><br />
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<i>n</i><sup>2</sup> <img src='images/symbol_times.gif' width='9' height='9' alt='×' border='0' style='vertical-align:middle;' /> <i>n</i> = <i>n</i><sup>3</sup><br />
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<i>n</i><sup>3</sup> <img src='images/symbol_times.gif' width='9' height='9' alt='×' border='0' style='vertical-align:middle;' /> <i>n</i><sup>3</sup> = <i>n</i><sup>6</sup><br />
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<i>n</i><sup>6</sup> <img src='images/symbol_times.gif' width='9' height='9' alt='×' border='0' style='vertical-align:middle;' /> <i>n</i><sup>6</sup> = <i>n</i><sup>12</sup><br />
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<i>n</i><sup>12</sup> <img src='images/symbol_times.gif' width='9' height='9' alt='×' border='0' style='vertical-align:middle;' /> <i>n</i><sup>3</sup> = <i>n</i><sup>15</sup></p>
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<p>We shall define m(<i>k</i>) to be the minimum number of multiplications to compute <i>n</i><sup><i>k</i></sup>; for example m(15) = 5.</p>
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<p>For 1 <img src='images/symbol_le.gif' width='10' height='12' alt='≤' border='0' style='vertical-align:middle;' /> <i>k</i> <img src='images/symbol_le.gif' width='10' height='12' alt='≤' border='0' style='vertical-align:middle;' /> 200, find <span style='font-family:times new roman;font-size:13pt;'><img src='images/symbol_sum.gif' width='11' height='14' alt='∑' border='0' style='vertical-align:middle;' /></span> m(<i>k</i>).</p>
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