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---
id: 5900f4301000cf542c50ff42
title: 'Problem 196: Prime triplets'
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challengeType: 5
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forumTopicId: 301834
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dashedName: problem-196-prime-triplets
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---
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# --description--
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Build a triangle from all positive integers in the following way:
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$$\begin{array}{rrr}
& 1 \\\\
& \color{red}{2} & \color{red}{3} \\\\
& 4 & \color{red}{5} & 6 \\\\
& \color{red}{7} & 8 & 9 & 10 \\\\
& \color{red}{11} & 12 & \color{red}{13} & 14 & 15 \\\\
& 16 & \color{red}{17} & 18 & \color{red}{19} & 20 & 21 \\\\
& 22 & \color{red}{23} & 24 & 25 & 26 & 27 & 28 \\\\
& \color{red}{29} & 30 & \color{red}{31} & 32 & 33 & 34 & 35 & 36 \\\\
& \color{red}{37} & 38 & 39 & 40 & \color{red}{41} & 42 & \color{red}{43} & 44 & 45 \\\\
& 46 & \color{red}{47} & 48 & 49 & 50 & 51 & 52 & \color{red}{53} & 54 & 55 \\\\
& 56 & 57 & 58 & \color{red}{59} & 60 & \color{red}{61} & 62 & 63 & 64 & 65 & 66 \\\\
& \cdots
\end{array}$$
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Each positive integer has up to eight neighbours in the triangle.
A set of three primes is called a prime triplet if one of the three primes has the other two as neighbours in the triangle.
For example, in the second row, the prime numbers 2 and 3 are elements of some prime triplet.
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If row 8 is considered, it contains two primes which are elements of some prime triplet, i.e. 29 and 31. If row 9 is considered, it contains only one prime which is an element of some prime triplet: 37.
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Define $S(n)$ as the sum of the primes in row $n$ which are elements of any prime triplet. Then $S(8) = 60$ and $S(9) = 37$.
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You are given that $S(10000) = 950007619$.
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Find $S(5678027) + S(7208785)$.
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# --hints--
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`primeTriplets()` should return `322303240771079940` .
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```js
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assert.strictEqual(primeTriplets(), 322303240771079940);
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```
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# --seed--
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## --seed-contents--
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```js
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function primeTriplets() {
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return true;
}
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primeTriplets();
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```
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# --solutions--
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```js
// solution required
```