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<meta name="author" content="Colin Hughes" />
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<title>Problem 196 - Project Euler</title>
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<h2>Prime triplets</h2><div class="info" style="cursor:help;width:200px;margin-bottom:10px;"><h3>Problem 196</h3><span style="width:300px;color:#666;">Published on Friday, 30th May 2008, 06:00 pm; Solved by 1245</span></div>
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<div class="problem_content" role="problem">
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<p>Build a triangle from all positive integers in the following way:</p>
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<p style="font-family:courier new,monospace;font-weight:bold;margin-left:50px;"> 1<br />
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<span style="color:red;">2</span> <span style="color:red;">3</span><br />
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4 <span style="color:red;">5</span> 6<br />
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<span style="color:red;">7</span> 8 9 10<br />
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<span style="color:red;">11</span> 12 <span style="color:red;">13</span> 14 15<br />
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16 <span style="color:red;">17</span> 18 <span style="color:red;">19</span> 20 21<br />
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22 <span style="color:red;">23</span> 24 25 26 27 28<br />
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<span style="color:red;">29</span> 30 <span style="color:red;">31</span> 32 33 34 35 36<br />
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<span style="color:red;">37</span> 38 39 40 <span style="color:red;">41</span> 42 <span style="color:red;">43</span> 44 45<br />
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46 <span style="color:red;">47</span> 48 49 50 51 52 <span style="color:red;">53</span> 54 55<br />
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56 57 58 <span style="color:red;">59</span> 60 <span style="color:red;">61</span> 62 63 64 65 66<br />
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. . .</p>
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<p>Each positive integer has up to eight neighbours in the triangle.</p>
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<p>A set of three primes is called a <i>prime triplet</i> if one of the three primes has the other two as neighbours in the triangle.</p>
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<p>For example, in the second row, the prime numbers 2 and 3 are elements of some prime triplet.</p>
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<p>If row 8 is considered, it contains two primes which are elements of some prime triplet, i.e. 29 and 31.<br />
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If row 9 is considered, it contains only one prime which is an element of some prime triplet: 37.</p>
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<p>Define S(<var>n</var>) as the sum of the primes in row <var>n</var> which are elements of any prime triplet.<br />
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Then S(8)=60 and S(9)=37.</p>
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<p>You are given that S(10000)=950007619.</p>
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<p>Find S(5678027) + S(7208785).</p>
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