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project_euler/pe/p318.html
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project_euler/pe/p318.html
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<!DOCTYPE html>
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<html lang="en">
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<head>
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<meta charset="utf-8" />
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<meta name="author" content="Colin Hughes" />
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<title>Problem 318 - Project Euler</title>
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<h2>2011 nines</h2><div class="info" style="cursor:help;width:200px;margin-bottom:10px;"><h3>Problem 318</h3><span style="width:300px;color:#666;">Published on Saturday, 1st January 2011, 04:00 pm; Solved by 483</span></div>
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<div class="problem_content" role="problem">
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<p>
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Consider the real number <img src='images/symbol_radic.gif' width='14' height='16' alt='√' border='0' style='vertical-align:middle;' />2+<img src='images/symbol_radic.gif' width='14' height='16' alt='√' border='0' style='vertical-align:middle;' />3.<BR />
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When we calculate the even powers of <img src='images/symbol_radic.gif' width='14' height='16' alt='√' border='0' style='vertical-align:middle;' />2+<img src='images/symbol_radic.gif' width='14' height='16' alt='√' border='0' style='vertical-align:middle;' />3
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we get:<BR />
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(<img src='images/symbol_radic.gif' width='14' height='16' alt='√' border='0' style='vertical-align:middle;' />2+<img src='images/symbol_radic.gif' width='14' height='16' alt='√' border='0' style='vertical-align:middle;' />3)<sup>2</sup> = 9.898979485566356...<BR />
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(<img src='images/symbol_radic.gif' width='14' height='16' alt='√' border='0' style='vertical-align:middle;' />2+<img src='images/symbol_radic.gif' width='14' height='16' alt='√' border='0' style='vertical-align:middle;' />3)<sup>4</sup> = 97.98979485566356...<BR />
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(<img src='images/symbol_radic.gif' width='14' height='16' alt='√' border='0' style='vertical-align:middle;' />2+<img src='images/symbol_radic.gif' width='14' height='16' alt='√' border='0' style='vertical-align:middle;' />3)<sup>6</sup> = 969.998969071069263...<BR />
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(<img src='images/symbol_radic.gif' width='14' height='16' alt='√' border='0' style='vertical-align:middle;' />2+<img src='images/symbol_radic.gif' width='14' height='16' alt='√' border='0' style='vertical-align:middle;' />3)<sup>8</sup> = 9601.99989585502907...<BR />
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(<img src='images/symbol_radic.gif' width='14' height='16' alt='√' border='0' style='vertical-align:middle;' />2+<img src='images/symbol_radic.gif' width='14' height='16' alt='√' border='0' style='vertical-align:middle;' />3)<sup>10</sup> = 95049.999989479221...<BR />
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(<img src='images/symbol_radic.gif' width='14' height='16' alt='√' border='0' style='vertical-align:middle;' />2+<img src='images/symbol_radic.gif' width='14' height='16' alt='√' border='0' style='vertical-align:middle;' />3)<sup>12</sup> = 940897.9999989371855...<BR />
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(<img src='images/symbol_radic.gif' width='14' height='16' alt='√' border='0' style='vertical-align:middle;' />2+<img src='images/symbol_radic.gif' width='14' height='16' alt='√' border='0' style='vertical-align:middle;' />3)<sup>14</sup> = 9313929.99999989263...<BR />
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(<img src='images/symbol_radic.gif' width='14' height='16' alt='√' border='0' style='vertical-align:middle;' />2+<img src='images/symbol_radic.gif' width='14' height='16' alt='√' border='0' style='vertical-align:middle;' />3)<sup>16</sup> = 92198401.99999998915...<BR />
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</p>
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<p>
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It looks like that the number of consecutive nines at the beginning of the fractional part of these powers is non-decreasing.<BR />
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In fact it can be proven that the fractional part of (<img src='images/symbol_radic.gif' width='14' height='16' alt='√' border='0' style='vertical-align:middle;' />2+<img src='images/symbol_radic.gif' width='14' height='16' alt='√' border='0' style='vertical-align:middle;' />3)<sup>2n</sup> approaches 1 for large n.
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</p>
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<p>
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Consider all real numbers of the form <img src='images/symbol_radic.gif' width='14' height='16' alt='√' border='0' style='vertical-align:middle;' />p+<img src='images/symbol_radic.gif' width='14' height='16' alt='√' border='0' style='vertical-align:middle;' />q with p and q positive integers and p<img src='images/symbol_lt.gif' width='10' height='10' alt='<' border='0' style='vertical-align:middle;' />q, such that the fractional part
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of (<img src='images/symbol_radic.gif' width='14' height='16' alt='√' border='0' style='vertical-align:middle;' />p+<img src='images/symbol_radic.gif' width='14' height='16' alt='√' border='0' style='vertical-align:middle;' />q)<sup>2n</sup> approaches 1 for large n.
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</p>
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<p>
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Let C(p,q,n) be the number of consecutive nines at the beginning of the fractional part of <BR /> (<img src='images/symbol_radic.gif' width='14' height='16' alt='√' border='0' style='vertical-align:middle;' />p+<img src='images/symbol_radic.gif' width='14' height='16' alt='√' border='0' style='vertical-align:middle;' />q)<sup>2n</sup>.
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</p>
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<p>
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Let N(p,q) be the minimal value of n such that C(p,q,n) <img src='images/symbol_ge.gif' width='10' height='12' alt='≥' border='0' style='vertical-align:middle;' /> 2011.
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</p>
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<p>
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Find <img src='images/symbol_sum.gif' width='11' height='14' alt='∑' border='0' style='vertical-align:middle;' />N(p,q) for p+q <img src='images/symbol_le.gif' width='10' height='12' alt='≤' border='0' style='vertical-align:middle;' /> 2011.
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</p>
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