108 lines
6.5 KiB
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108 lines
6.5 KiB
HTML
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<h2>Diophantine reciprocals I</h2><div class="info" style="cursor:help;width:200px;margin-bottom:10px;"><h3>Problem 108</h3><span style="width:300px;color:#666;">Published on Friday, 4th November 2005, 06:00 pm; Solved by 6511</span></div>
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<p>In the following equation <var>x</var>, <var>y</var>, and <var>n</var> are positive integers.</p>
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<div style="text-align:center;">
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<table align="center">
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<tr>
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<td><div style="text-align:center;">1<br /><img src="images/blackdot.gif" width="15" height="1" alt="" /><br /> <var>x</var></div></td>
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<td> + </td>
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<td><div style="text-align:center;">1<br /><img src="images/blackdot.gif" width="15" height="1" alt="" /><br /> <var>y</var></div></td>
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<td> = </td>
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<td><div style="text-align:center;">1<br /><img src="images/blackdot.gif" width="15" height="1" alt="" /><br /> <var>n</var></div></td>
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</tr>
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</table>
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<p>For <var>n</var> = 4 there are exactly three distinct solutions:</p>
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<div style="text-align:center;">
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<table align="center">
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<tr>
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<td><div style="text-align:center;">1<br /><img src="images/blackdot.gif" width="15" height="1" alt="" /><br /> 5</div></td>
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<td> + </td>
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<td><div style="text-align:center;">1<br /><img src="images/blackdot.gif" width="15" height="1" alt="" /><br /> 20</div></td>
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<td> = </td>
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<td><div style="text-align:center;">1<br /><img src="images/blackdot.gif" width="15" height="1" alt="" /><br /> 4</div></td>
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</tr>
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<tr>
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<td><div style="text-align:center;">1<br /><img src="images/blackdot.gif" width="15" height="1" alt="" /><br /> 6</div></td>
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<td> + </td>
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<td><div style="text-align:center;">1<br /><img src="images/blackdot.gif" width="15" height="1" alt="" /><br /> 12</div></td>
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<td> = </td>
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<td><div style="text-align:center;">1<br /><img src="images/blackdot.gif" width="15" height="1" alt="" /><br /> 4</div></td>
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</tr>
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<tr>
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<td><div style="text-align:center;">1<br /><img src="images/blackdot.gif" width="15" height="1" alt="" /><br /> 8</div></td>
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<td> + </td>
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<td><div style="text-align:center;">1<br /><img src="images/blackdot.gif" width="15" height="1" alt="" /><br /> 8</div></td>
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<td> = </td>
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<td><div style="text-align:center;">1<br /><img src="images/blackdot.gif" width="15" height="1" alt="" /><br /> 4</div></td>
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</tr>
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</table>
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<p>What is the least value of <var>n</var> for which the number of distinct solutions exceeds one-thousand?</p>
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<p class="info">NOTE: This problem is an easier version of problem <a href="index.php?section=problems&id=110">110</a>; it is strongly advised that you solve this one first.</p>
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