146 lines
6.2 KiB
HTML
146 lines
6.2 KiB
HTML
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<h2>Primes with runs</h2><div class="info" style="cursor:help;width:200px;margin-bottom:10px;"><h3>Problem 111</h3><span style="width:300px;color:#666;">Published on Friday, 16th December 2005, 06:00 pm; Solved by 3437</span></div>
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<div class="problem_content" role="problem">
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<p>Considering 4-digit primes containing repeated digits it is clear that they cannot all be the same: 1111 is divisible by 11, 2222 is divisible by 22, and so on. But there are nine 4-digit primes containing three ones:</p>
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<p style='text-align:center;'>1117, 1151, 1171, 1181, 1511, 1811, 2111, 4111, 8111</p>
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<p>We shall say that M(<i>n</i>, <i>d</i>) represents the maximum number of repeated digits for an <i>n</i>-digit prime where <i>d</i> is the repeated digit, N(<i>n</i>, <i>d</i>) represents the number of such primes, and S(<i>n</i>, <i>d</i>) represents the sum of these primes.</p>
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<p>So M(4, 1) = 3 is the maximum number of repeated digits for a 4-digit prime where one is the repeated digit, there are N(4, 1) = 9 such primes, and the sum of these primes is S(4, 1) = 22275. It turns out that for <i>d</i> = 0, it is only possible to have M(4, 0) = 2 repeated digits, but there are N(4, 0) = 13 such cases.</p>
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<p>In the same way we obtain the following results for 4-digit primes.</p>
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<div style='text-align:center;'>
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<table align='center' border='1' cellspacing='0' cellpadding='5'>
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<tr>
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<td><b>Digit, <i>d</i></b></td>
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<td><b>M(4, <i>d</i>)</b></td>
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<td><b>N(4, <i>d</i>)</b></td>
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<td><b>S(4, <i>d</i>)</b></td>
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</tr>
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<tr>
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<td>0</td>
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<td>2</td>
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<td>13</td>
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<td>67061</td>
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</tr>
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<tr>
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<td>1</td>
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<td>3</td>
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<td>9</td>
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<td>22275</td>
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</tr>
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<tr>
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<td>2</td>
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<td>3</td>
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<td>1</td>
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<td>2221</td>
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</tr>
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<tr>
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<td>3</td>
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<td>3</td>
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<td>12</td>
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<td>46214</td>
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</tr>
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<tr>
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<td>4</td>
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<td>3</td>
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<td>2</td>
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<td>8888</td>
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</tr>
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<tr>
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<td>5</td>
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<td>3</td>
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<td>1</td>
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<td>5557</td>
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</tr>
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<tr>
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<td>6</td>
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<td>3</td>
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<td>1</td>
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<td>6661</td>
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</tr>
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<tr>
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<td>7</td>
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<td>3</td>
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<td>9</td>
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<td>57863</td>
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</tr>
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<tr>
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<td>8</td>
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<td>3</td>
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<td>1</td>
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<td>8887</td>
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</tr>
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<tr>
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<td>9</td>
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<td>3</td>
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<td>7</td>
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<td>48073</td>
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</tr>
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</table>
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</div>
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<p>For <i>d</i> = 0 to 9, the sum of all S(4, <i>d</i>) is 273700.</p>
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<p>Find the sum of all S(10, <i>d</i>).</p>
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