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puzzles/project_euler/pe/p066.html
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puzzles/project_euler/pe/p066.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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<meta name="keywords" content="programming,mathematics,problems,puzzles" />
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<title>Problem 66 - Project Euler</title>
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<h2>Diophantine equation</h2><div class="info" style="cursor:help;width:200px;margin-bottom:10px;"><h3>Problem 66</h3><span style="width:300px;color:#666;">Published on Friday, 26th March 2004, 06:00 pm; Solved by 8716</span></div>
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<div class="problem_content" role="problem">
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<p>Consider quadratic Diophantine equations of the form:</p>
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<p style='text-align:center;'><i>x</i><sup>2</sup> – D<i>y</i><sup>2</sup> = 1</p>
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<p>For example, when D=13, the minimal solution in <i>x</i> is 649<sup>2</sup> – 13<img src='images/symbol_times.gif' width='9' height='9' alt='×' border='0' style='vertical-align:middle;' />180<sup>2</sup> = 1.</p>
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<p>It can be assumed that there are no solutions in positive integers when D is square.</p>
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<p>By finding minimal solutions in <i>x</i> for D = {2, 3, 5, 6, 7}, we obtain the following:</p>
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<p style='margin-left:20px;'>3<sup>2</sup> – 2<img src='images/symbol_times.gif' width='9' height='9' alt='×' border='0' style='vertical-align:middle;' />2<sup>2</sup> = 1<br />
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2<sup>2</sup> – 3<img src='images/symbol_times.gif' width='9' height='9' alt='×' border='0' style='vertical-align:middle;' />1<sup>2</sup> = 1<br />
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<span style='color:#dd0000;font-weight:bold;'>9</span><sup>2</sup> – 5<img src='images/symbol_times.gif' width='9' height='9' alt='×' border='0' style='vertical-align:middle;' />4<sup>2</sup> = 1<br />
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5<sup>2</sup> – 6<img src='images/symbol_times.gif' width='9' height='9' alt='×' border='0' style='vertical-align:middle;' />2<sup>2</sup> = 1<br />
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8<sup>2</sup> – 7<img src='images/symbol_times.gif' width='9' height='9' alt='×' border='0' style='vertical-align:middle;' />3<sup>2</sup> = 1</p>
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<p>Hence, by considering minimal solutions in <i>x</i> for D <img src='images/symbol_le.gif' width='10' height='12' alt='≤' border='0' style='vertical-align:middle;' /> 7, the largest <i>x</i> is obtained when D=5.</p>
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<p>Find the value of D <img src='images/symbol_le.gif' width='10' height='12' alt='≤' border='0' style='vertical-align:middle;' /> 1000 in minimal solutions of <i>x</i> for which the largest value of <i>x</i> is obtained.</p>
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